Hi, I am fairly new to working with an arduino and the whole EE field in general.
Are you a good programmer? I'd say the software is the tough part. 
Just for reference, neither of those projects looks like a spectrum analyzer, although there is apparently some frequency analysis going on... That's OK... Whatever effect you want!
What I figured i need is an external power supply to power about 50 ws2811 leds
For a large number of LEDs, you'll need an external power supply and some kind of driver circuit.
With a [u]serial shift-register approach[/u] you can address an almost unlimited number of LEDs with just 3 output lines. Another approach is a matrix connection. (A matrix LED arrangement does not require a matrix connection.) There are special purpose driver chips, and you can look around. I've got a lighting effect with 48 individually-addressable LEDs, controlled serially with six [u]MAX6968[/u] chips.
and a pair of old headphones to transmit audio to the arduino but I do not know the in between aka what the breadboard and arduino should look like
What? Headphones don't transmit audio... A headphone-output from a laptop or smartphone, etc., can work if the volume is cranked-up. A line-level signal (the RCA connections from a CD or DVD player, etc.) is also good.
However, the Arduino cannot accept the negative voltage swing of an audio signal. The easiest solution is to [u]bias the input[/u] at 2.5V. Then you simply subtract-out the bias in software.
If you don't need more than 7 frequency bands, check-out the [u]MSGEQ7 chip[/u]. It has 7 bandpass filters built-in and a single time-multiplexed variable DC output. The chip also takes care of the negative voltage issue. It's not super-easy to program (because the timing is critical) but I'd say it's 10 times easier than FFT or software-digital filtering.
[u]This is a spectrum analyzer[/u]. A spectrum analyzer (of this type) is basically a bunch bargraph-type meters, each meter representing a different band of frequencies. Traditionally, the low frequencies are on the left and high frequencies on the right.
A real spectrum analyzer is an instrument with calibrated & labeled frequency & dB scales, but a spectrum analyzer effect doesn't have to be precise. i.e. I've made a VU Meter effect that automatically adjusts/calibrates to the average level so it's totally useless as any kind of meter, but as a meter effect it automatically adjusts for quiet songs & loud songs or volume changes and you always get a lot of "meter action".
