A way to directly see that the interior angles of triangle sum to $180^\circ$?

Since that fact about the angle sum is equivalent to the parallel postulate, any visualization is likely to include a pair of parallel lines. Here's one from wikipedia:

Ethan Bolker's answer is the standard proof, but a more visual way to see the result is to tile the plane with copies of your triangle. It is so effective, I will not even include a drawing.

Just think about it... the tiling is made by three sets of parallel lines in the directions of the triangle sides, and they meet at vertices of the tiling, where you will then find two copies of each angle.

Edit. By popular request, there's a picture below:

Edit 2. I like Steven Gubkin's answer better :) That is what I used to call the "near-sighted method"

This is very similar to Ethan Bolker's and Rodrigo A. Pérez's answers, but I made a small animation to illustrate a version that I like.