There are three steps to doing what you want, and the Arduino is going to struggle with all of them.
First you need to sample a short block of audio into a buffer that is a power-of-two in size with at least double the number of samples that you want to have bars, and at double the frequency of the highest frequency band you are interested in.
Once you have that sample set you then need to perform a Fast Fourier Transform (FFT) on it. This is quite a heavy mathematical operation, but there are integer only implementations of it that work kind of OK on a little Arduino (though a floating point version running on a better chip with FPU, or a DSP version on a chip with a DSP, is more desirable).
Then the third step is to take the results of the FFT (or "buckets") and calculate the power magnitude of each one (basically the square root of the sum of the squares of the real and imaginary values of the bucket - just like Pythagoras with a triangle).
Finally you can plot those power values on your LED bars.