Why are projective spaces and varieties preferable?

Imagine some student comes and says to you "Why do we have to work with complex numbers? It's really much easier to draw functions from $\mathbb R \to \mathbb R$" and for me real numbers are more intuitive. What would you answer? If I was this student, a motivating answer for me would be something like "The equation $x^2 + 1 = 0$ shows to us that if we want to solve algebraic equations, we need to work with $\mathbb C$ and more generally with algebraically closed fields." Nustellensatz doesn't work in non-algebraically closed fields.

Now back to your problem, do you know Bézout Theorem ?

If $\mathcal C_1, \mathcal C_2$ are two curves of degree $d,e$ in $\mathbb P^2$, then they have $d\cdot e$ points of intersection (counted with multiplicities).

Bézout's theorem is obviously false for $\mathbb A^2$ : take two parallel lines. Projective space allowed us to simplify lot of situations, and is the good framework if you want to do interesection theory for example. Homogenous coordinates are incredibly elegant and useful!. If you need a motivation to work only in projective space, you can take a look at the elegant book of Benoit Kloeckner (in French) or the book of Pierre Samuel.

If you need more motivating examples, "Un bref aperçu de la géométrie projective" (but I'm sure plenty of other books do actually contain the same stuff) gives for several theorems (Pappos for sure and maybe Pascal) a first affine proof and after that a shortened and more elegant projective proof. I think it's very instructive to read a bit about it !

Looking at problems in the projective setting is a kind of compactification. Compactness is a strong property which makes life easy lots of the time.

I will make this a community wiki answer and start a list of statements that make working with projective varieties particularly interesting:

• The image of a morphism $f:X\to Y$ is closed if $X$ is projective. In general, the image of a morphism of varieties is not a variety.
• As N.H. already mentioned, intersections in projective space behave "well" in the sense of Bézout's Theorem.