I feel that the thing to do is to start off by thinking about real projective spaces. There are also some tricks to help understand projective spaces in general.
One trick is to "lift" back to the underlying vector space. For example, to come to grips with the idea that any two projective lines in $\mathbb P^2(\mathbb R)$ meet in one point, we can think about what happens in the vector space $\mathbb R^3.$ A projective line is a linear collection of projective points, which themselves are described by lines through the origin in $\mathbb R^3.$ So, a projective line is described by a plane through the origin in $\mathbb R^3.$ Translating what "two lines in $\mathbb P^2(\mathbb R)$ meet in a point" means to their representatives in $\mathbb R^3,$ the statement becomes that any two planes through the origin intersect in a unique line. Can you visualize this?
Another trick is to remember that projective space decomposes (non-uniquely) into disjoint pieces: $\mathbb P^n = \mathbb A^n \sqcup\mathbb P^{n-1}.$ Often times, we can ignore the fact that we're working with projective space, if the behaviour we're interested in takes place entirely within the $\mathbb A^n$ piece.
For example, the projective plane is what we get by adding the projective line to the affine plane $\mathbb A^2$ (in our case $\mathbb A^2 = \mathbb R^2,$ but without the vector space "structure", i.e., we don't add points of affine space, or multiply them by scalars). Remember from your notes, the projective line $\mathbb P^1(\mathbb R)$ can be thought of as the circle $S^1$ with antipodal points identified. If we put $S^1$ inside $\mathbb R^2$ as the unit circle centred at the origin, then every line through the origin defines a unique point of $\mathbb P^1,$ by its intersection with $S^1.$ This means that the projective plane is what we get by adding to the affine plane one point for each direction (i.e. slope!) that a line can have in $\mathbb R^2.$ If we are interested in the intersection of two lines in $\mathbb R^2$ that we know have different slopes, then we can ignore projective geometry. But, if two lines are parallel, then they do not intersect in $\mathbb R^2,$ but they do define the same point of $\mathbb P^1,$ which means that they intersect in the $\mathbb P^1$ portion of $\mathbb P^2$ ("at infinity").
Regarding visualizing complex geometry as real geometry, well, it's limited, because visualizing geometry past 4 dimensions isn't really possible. So, as soon as we think about spaces of complex dimension two or more, we're stuck. I think it's better to understand the real picture where we can, and then try to accept that the complex numbers arise in a way that preserves (and simplifies) the essential algebraic properties of the real numbers. When you study algebraic geometry, you will learn that geometric properties (in the situations we can visualize) can be expressed by algebraic conditions (vanishing of polynomials, etc.). The algebraic conditions can make perfect sense over different fields and in any dimension, but you will still only be able to visualize the geometry in $\mathbb R^1,\mathbb R^2,\mathbb R^3$ (maybe also over finite fields...). This doesn't mean we cannot draw diagrams to help our understanding, but those diagrams will no longer be so closely linked to the literal solutions to our equations, if that makes sense. But it's surprising how far one can get with "meaningless" pictures, so I encourage you to draw them in the real case, and to keep them in mind over other fields!
Anyway, those notes you are reading look very nice, I would certainly recommend you to keep reading them, and things will become clearer the more you do.
...Are there any suggestions as to resources I can look at for efficiently gaining a solidly intuitive, but not necessarily deep, understanding of differential geometry specifically for the purpose of motivating related ideas in algebraic geometry?...
If one woulds like to develop the intuition in differential geometry, I suggest:
- do Carmo M. P. - Differential Geometry of Curves and Surfaces;
- Spivak M. - A comprehensive introduction to differential geometry;
Volume 1; Volume 2, chapters 1, 2 and 3; Volume 3, chapters 2 and 3;
- Wells R. O. Jr. and Garcia-Prada O. - Differential Analysis on Complex Manifolds, chapters 1, 2, 3 and 5.
If one woulds like to develop the intuition in algebraic geometry, I suggest:
- Eisenbud D., Harris J. - The Geometry of Schemes, the chapter 2;
- Fischer G. - Plane Algebraic Curves;
- Hulek K. - Elementary Algebraic Geometry;
- Mumford D. - The Red Book of Varieties and Schemes.
And, as recap in the complex differential and algebraic frameworks, I suggest:
- Griffiths P. and Harris J. - Principles of Algebraic Geometry, chapters 0, 1 and 2;
- Neeman A. - Algebraic and Analytic Geometry;
- Voisin C. - Hodge Theory and Complex Algebraic Geometry, the first parts of volumes I and II.
Best Answer
The most appropriate answer will depend on why you are working through a book on Riemann surfaces and algebraic curves, but I will try to give some suggestions.
Since you mention Riemann surfaces, let's start with some analogy with smooth manifolds. The Whitney embedding theorem says that any smooth manifold can be embedded in $\mathbb{R}^N$ for $N$ sufficiently large, so we can always think of a smooth manifold as a submanifold of $\mathbb{R}^N$. This occasionally helps with intuition and visualization, and can simplify some constructions.
In the case of complex manifolds (e.g. Riemann surfaces), you might ask whether the same holds true holomorphically, i.e. whether any complex manifold can be holomorphically embedded in $\mathbb{C}^N$ for $N$ sufficiently large. It turns out that usually the answer is no. It is an easy consequence of the Liouville theorem that no compact complex manifold is a complex submanifold of $\mathbb{C}^N$. If you only care about compact complex manifolds, then $\mathbb{CP}^N$ turns out to be the best possible (see e.g. the Kodaira embedding theorem, which characterizes which compact complex manifolds are complex submanifolds of $\mathbb{CP}^N$).
If your motivation is the study of solutions to polynomial equations, then as mentioned in other answers and comments, projective spaces are the appropriate completions of affine space that allow as many solutions as possible, allowing various formulas (e.g. couting intersections) work without additional qualification.
About visualization: for curves in $\mathbb{CP}^2$, first take some affine chart $\mathbb{C}^2 \subset \mathbb{CP}^2$, and then look at the intersection with some "real slice" $\mathbb{R}^2 \subset \mathbb{C}^2$. For example if we look at the curve in $\mathbb{CP}^2$ given by the zero set of $x^2-yz$, by working on the affine chart $z\neq0$ this becomes $y = x^2$ on $\mathbb{C}^2$, and if we restrict to real $x,y$ we get a parabola.