There are ready-made devices that can do this, or you can build it yourself. You're correct in thinking that some kind of amplifier (or switch, more properly) is needed; the headphone signal isn't strong enough to light LEDs directly.
What electronic skills do you have?
- Being able to read a schematic (this is essential, and not rocket science)
- Soldering or breadboarding
- Looking up stuff on-line or in the library
People here will probably be glad to help you come up with a simple DIY "color organ" (that's basically what you're asking about here).
It would use the electrical signal to tell the LEDs to light up on an external power source, like a small battery of some sort.