that's the digital version of the old 'color organ'. Design wise, a simple AC colour organ (as in these kits
or this schematic
) will be much easier to realize, cheaper and much brighter!
You would need thousands of dollars of LED's to make something as bright as a $20 colour organ. And that is only a very slight exaggeration.
Doing it on an Arduino will require some audio filters for the input, to 'select' audio within a certain frequency range, filter this into DC and pass it to an analog input. For design simplicity you should probably look for a passive R/C filter-- just google those terms. It's really unlikely that you would need an LM386, as the design problem here is not to amplify thw audio signal to higher power level, but rather to break the audio down into separate low-power signals that represent each frequency band.
If you do end up needing an op-amp, it will be the low-power instrumentation variety, like a TL 072... with some resistors and capacitors, the op amp can become an 'active filter' that can pick out any particular range of frequencies you like.