I will focus on the representation theory of the objects involved, which is a place where the quote mentioned is pretty much precisely what happens.
So let's start with an algebraic group $G$ over $\mathbb{C}$. We would like to know something about the representations of this group (by representation I mean rational representation).
Let's also assume that we are very good at this whole business with representations of Lie algebras, so if we could somehow get a Lie algebra from our algebraic group, that would be good. If this Lie algebra can somehow tell us something about the representations of the algebraic group, then that will be even better.
Fortunately, this is exactly what we can do. Associated to $G$ is a Lie algebra $\operatorname{Lie}(G)$, and the representations of this Lie algebra correspond directly to those of $G$ (I will make this more precise a bit later).
Before I go into further details about this Lie algebra and how the representations relate to each other, I would like to take a brief detour through a more general case, namely that where we work over an algebraically closed field whose characteristic need not be $0$ (to do this properly, one should then have $G$ be a group scheme, but I will not get into that here).
When the characteristic is not $0$, we still get a Lie algebra, but this Lie algebra is "too small" to capture the representations of the group (instead, it is a restricted Lie algebra, and its restricted representations correspond to the representations of the first Frobenius kernel of $G$. Ignore the previous sentence if you are not familiar with schemes).
So is there something we can do that works in all characteristics?
The answer is yes, and it is called the algebra of distributions on $G$ (or sometimes the hyperalgebra of $G$). I will not go into how this is constructed here, but just note that no matter the characteristic, we have an equivalence of categories between the representations of the algebra of distributions on $G$ and the representations of $G$.
Finally, how does this algebra of distributions on $G$ relate to the Lie algebra of $G$?
Here, the answer is very nice in characteristic $0$, since the algebra of distributions on $G$ is in fact isomorphic to the universal enveloping algebra of $\operatorname{Lie}(G)$, and hence representations of this algebra correspond precisely to representations of $\operatorname{Lie}(G)$.
The above of course lacks any sort of detail, but often it is good to see the big picture first, so that is what I decided to provide here.
For more details, the books called "Linear Algebraic Groups" are a common reference (there is one each by Borel, Humphreys and Springer, though I am not familiar with the Borel one myself). I am not sure if any of those actually discuss the algebra of distributions (as the main focus there is on characteristic $0$ where it is not really needed), so if you want a more advanced treatment, I recommend Representations of Algebraic Groups by Jantzen (but beware that this is a very advanced book that assumes a good familiarity with at least one of the previously mentioned books as well as some further algebraic geometry).
Best Answer
The representation theory of Lie groups and Lie algebras are very related. In fact, in the case of Simply-connected Lie groups, the irreducible representations of these Lie groups are in bijection with the irreducible representations of its corresponding Lie algebra. In the case of a connected Lie group, the irreducible representations of its corresponding Lie algebra are in bijection with the irreducible representations of its universal covering space ( which is a Lie group as well ). If your Lie group is not connected, one can still use this correspondence by considering the connected component of the identity ( your group modulo this component will only be a discrete/dimension 0 group ).
Off the top of my head, I can give 2 good books that illustrate this correspondence very well: Representation theory: a first course (Fulton and Harris) and Introduction to Lie Groups and Lie algebras ( Krilliov Jr ). You can also learn a good deal by reading the first couple of chapters of Representations of Compact Lie groups by Bockner. However, the rest of the book develops the theory almost entirely without any reference to Lie algebras.