How new are you to EE?
The block diagram isn't too bad. You take a signal and run it into several parallels circuits. Each circuit has steep filters to narrow down the specific frequency range for that circuit. The amplitude of that resulting output on that range is then converted into a linear display format like a row of LEDs.
I own an AudioContol 3050A spectrum analyzer, and when opened its pretty much that, 31 circuits side by side to drive the display.
Nowadays, you can get PC software based analyzers that use FFTs to create an output with a whole lot less work....
If you really wanted to build it, you could start by looking at circuits for a VU meter, then how to add filters to it. Here's a couple commerical units one similar and one that's DSP based: