Real algebraic geometry comes with its own set of methods. While keeping in mind the complex picture is sometimes useful (e.g. for any real algebraic variety X, the Smith-Thom inequality asserts that $b(X(\mathbb{R})) \leq b(X(\mathbb{C}))$, where $b(\cdot)$ denotes the sum of the topological Betti numbers with mod 2 coefficients), most of the technique used are either built from scratch or borrow from other areas, such as singularity theory or model theory.
The literature is a lot smaller for RAG than for traditional AG; the basic reference is the book by Bochnak, Coste and Roy (preferably the English-language edition which is more recent by more than 10 years, and has been greatly expanded). The book covers in particular the real spectrum, the transfer principle (which makes non-standard methods really easy), stratifications and Nash manifolds, among other topics. Michel Coste also has An Introduction to Semialgebraic Geometry available on his webpage a very short treatment of some basic results, enough to give you a first impression.
Other interesting books tend to be shorter and more focused than BCR, dealing with a specific aspect; e.g. Prestel's Positive polynomials. (dealing mostly with results such as Schmudgen's theorem), and Andradas-Brocker-Ruiz Constructible sets in real geometry (dealing mostly with the minimum number of inequalities required to define basic sets). The book by Benedetti and Risler is very interesting and concrete; I found some passages very useful and some results are hard to find in other books (the sections on additive complexity of polynomials are very thorough), but it is a bit scatterbrained for my taste.
As the name indicates, the book by Basu Pollack and Roy is entirely focused on the algorithmic aspects. It's a very good book, and you may still pick up some of the theory in there, but it does not sound like what you are after right now.
As for o-minimality, there again, Michel Coste's webpage contains an introduction that nicely complements van den Dries's book. I would hesitate to bundle o-minimality with real algebraic geometry. In some respects, the two domains are undoubtedly close cousins, and o-minimality can be seen as a wide-ranging generalization of real algebraic structures; on the other hand, each disciplines has also its own aspects and problems that do not translate all that well into the other.
I'm being verbose as usual. Still, I hope it helps.
As others have suggested, your friend is getting it backwards. He's like a hammer asking what a carpenter is useful for.
Given a field (of mathematics, say), there are typically some fields that are more structured than it and others that are less structured. In mathematics, people often say the more structured ones are 'harder', and the less structured are 'softer'. For instance, in increasing order of hardness, we have sets, topological spaces, topological manifolds, differential manifolds, complex manifolds, complex algebraic varieties, algebraic varieties over the rational numbers, integral algebraic varieties. These are in a linear order, but if you throw in other subjects, you'll get a non-linear one. (p-adic algebraic geometry and Riemannian geometry immediately come to mind.)
(I think Gromov has some remarks at the end of an ICM address where he talks about this and gives other examples. Also, don't confuse 'harder' and 'softer' in this sense with what they mean in the sciences, which is essentially 'more precise' and 'less precise'. For instance, in science people say that biology is softer than chemistry. In fact, the two meanings are opposites because in science, more structured objects are less amenable to a precise analysis. But this typically isn't the case in mathematics.)
Now given a subject S and a harder subject H, it's usually true that most objects in S don't admit the structure of an object in H. For instance, most topological manifolds don't admit a complex structure. On the other hand, for the objects of S that do admit such a structure, their theory from the point of view of H is typically much richer than that from the point of view of S. For instance, the study of Riemann surfaces as topological spaces is less rich than their study as complex manifolds. You might say that softer subjects are broad and flexible and harder ones are rich and rigid. Mathematicians tend to view subjects that are softer than their specialty as general nonsense, and harder ones as excessively particular.
This is not to say that a soft field is easier or less interesting than a harder one. Even if it is true that the directly analogous question in the soft subject is easier (e.g. classify Riemann surfaces topologically rather than holomorphically), it just means that the people in the soft subject can move on and study more sophisticated objects. So they just get stuck later rather than sooner. For instance, over the past 50 years, a big fraction of the best number theorists have been studying elliptic curves over number fields. Now elliptic curves over the complex numbers are much easier (I think there hasn't been much new since the 19th century), so the algebraic geometers just moved on to higher genus or higher dimension and are grappling with the issues there, issues that are way out of reach in the presence of arithmetic structure.
Now my main point here is that soft subjects were typically invented to break up the study of harder ones into smaller pieces. (This is surely something of a creation myth, but one with a fair amount of truth.) For instance, the real numbers were invented to break up the study of polynomial equations into two steps: when a polynomial has a real solution and when that real solution is rational. I know very little about modern analysis, but I think that much of it was invented to do the same with differential equations. You first find solutions in some soft sense and then see whether it comes from a solution in the harder sense of original interest.
So the role of soft subjects is to aid in the study of harder ones---people usually don't ask for applications of partial differential equations to the study of topological vector spaces, but it's considered a mark of respectability to ask for the opposite. Similarly, no one talks about applications of engineering to mathematics. Since algebraic geometry is at the hard end of the spectrum above, there aren't many fields in which it is natural to ask for applications. Number theory, or arithmetic algebraic geometry, is harder and of course there are zillions of applications there, but that's not what your friend wants. Just about all mathematicians work in a subject that is softer than some and harder than others (and if you include non-mathematical subjects, then all mathematicians do). That's all good---it takes a whole food chain to make an ecosystem. But it's backwards to ask about the nutritional value of something that typically eats you.
[This picture of mathematics is of course simplistic. There are examples of hard subjects with applications to softer ones. See Donu Arapura's answer, for example. There are also applications of arithmetic algebraic geometry to complex algebraic geometry. For instance, Grothendieck's proof of the Ax-Grothendieck theorem, or the proof of the decomposition theorem for perverse sheaves using the theory of weights and the Weil conjectures. But I think it's fair to say that such applications are the exception---and are prized because of it---rather than the rule.]
Best Answer
You do forget things you are not working on. Nothing can be done about it. I could read German easily by the end of 8th grade and now I can hardly spell "Entshuldigen Sie mir bitte". There are several math. papers I read as a student of which I remember next to nothing. The most frustrating and shameful thing is that I don't remember the details of my own papers written 20 years ago with a few exceptions. After age 40 I also started to lose the ability I always took for granted: to get to the board at any time and start lecturing on some subject in my field with full proofs without any preparation. Now I have to sit for half an hour and to prepare my lectures now and then (thanks God this concerns only advanced graduate courses yet). And I work as a professional mathematician in academia full time!
The only way to cope with this loss of memory I know is to do some reading on systematic basis. Of course, if you read one paper in algebraic geometry (or whatever else) a month (or even two months), you may not remember the exact content of all of them by the end of the year but, since all mathematicians in one field use pretty much the same tricks and draw from pretty much the same general knowledge, you'll keep the core things in your memory no matter what you read (provided it is not patented junk, of course) and this is about as much as you can hope for.
Relating abstract things to "real life stuff" (and vice versa) is automatic when you work as a mathematician. For me, the proof of the Chacon-Ornstein ergodic theorem is just a sandpile moving over a pit with the sand falling down after every shift. I often tell my students that every individual term in the sequence doesn't matter at all for the limit but somehow together they determine it like no individual human is of any real importance while together they keep this civilization running, etc. No special effort is needed here and, moreover, if the analogy is not natural but contrived, it'll not be helpful or memorable. The standard mnemonic techniques are pretty useless in math. IMHO (the famous "foil" rule for the multiplication of sums of two terms is inferior to the natural "pair each term in the first sum with each term in the second sum" and to the picture of a rectangle tiled with smaller rectangles, though, of course, the foil rule sounds way more sexy).
Since it is a "general" question, I suggest making it community wiki (and mark my answer as such).