My personal story with this question is that, sometime in 2007, I wanted to find a project for a student I was mentoring at RSI (a program for high school students which produces real research) and thought some variant of the question "how can you visualize all the different geometric structures on a topological torus (elliptic curve/$\mathbb{C}$)?" would be motivational. (In the end, he went with a different algebraic geometry project and it was a great success. He is on this site and can probably guess who he is.) I mentioned this to a friend of mine, who does symplectic geometry (!), who said "isn't tropical geometry the canonical answer to that question?"

I found out that this was so. Through tropical geometry, you can represent nonisomorphic genus-one tropical "curves" as visually distinct planar graphs. I actually taught a topics course on the subject to half a dozen undergrads and learned other great visualizations: for example, the tropical version of Bezout's theorem is proven by actually counting triangles in the Newton polygon. The degree-genus formula too.

I even found out that it was practical (how practical? See this paper by Eric Katz): you can deduce actual algebraic geometry theorems from it. Of course the enumerative results of Mikhalkin are the most spectacular examples, but I was pleased to work out a deduction of Bezout's theorem over the field of Puiseux series from the tropical theorem (and, if you like nonconstructive isomorphisms, you can get the theorem over $\mathbb{C}$).

I don't actually do any research in the area, but it's very appealing.