Most of those flashy 2D games are hardware-accelerated, which allows the developer to take full advantage of the graphics hardware, and apply nicer-looking effects without impacting performance that much. Geometry Wars was most likely made with 2D polygonal lines or sprites, with several shaders for effects.
Shaders can be thought of as special programs that run in the GPU of the video card, that do additional stuff with pixels and vertices that would be much faster than doing it on the CPU. To get an idea of what shaders could do, here are a few examples. The explanations are very trivial and I think are pretty good for a total noob to start playing around with them. With some basic math and shader programming it's possible to get those "warp" or "glowing" effects you see in the background.
Object transformations are usually done with matrix-vector operations, or sometimes using linear algebra.
However, the graphics in that game are also simple to the point that a lot of those objects can be also created as sprites, and then it's a simple matter of blending sprites appropriately to create those effects. If you can reproduce the same visual effects from an artist's point of view, it's usually easier and faster to work that way.
I don't know of any good mobile frameworks, not having programmed for mobile platforms before. However I know that cocos2d is a good one, usually used for iOS games and it's based on OpenGL so it's hardware accelerated.