They are much closer to the curves and solids we're familiar with from everyday experience. For example: you have well-defined tangents at every point for a smooth curve, but a self-intersecting curve doesn't. So many techniques and a lot of intuition carries over into the abstract algebraic setting.

More generally, algebraic geometry is a central field of math because it is, basically, about solving polynomial equations. It's one of the most "well-connected" fields of mathematics today. Number theory, differential geometry, complex geometry, even differential equations: all these fields benefit from their interactions with modern algebraic geometry. (Also biology, I've heard. Lior Pachter is a name I remember in this connection.)