What exactly does an amplifier help with?

an amp will increase volume compared to, say, the power you get from an iPod, sure. But that’s not the main reason to get one.

 

The main benefits are the increased detail, more controlled and musical bass, better soundstage, better dynamic range and sense of attack for percussion.

 

so, basically, they can make everything better. :)

It's control more than anything.

Headphone drivers move back and forth. A good amp will precisely stop and start the driver. Poor or underpowered ones will let the cone keep moving forward or back when it should have stopped and gone in the other direction. That cleans up bass and gives a lot more clarity.

Like everyone said, the biggest difference I noticed from amping was a wider soundstage and much more crisp, refined timbre on the bass. It also made it much easier for the drivers to hit the low bass notes. 

An amp is also like a filter in a way. A good quality amp can take away unwanted harsh sounds, muddiness and other yucky things. 

 

I also find that a good amp can thicken the tone too and make a piano sound more like a piano.

Yeah I definitely agree on the piano. Sounds so much more real when amped.