In essence, a spectrum analyser boils down to a receiver with a swept oscillator with its output connected to a display such as an oscilloscope. In its most simple form it could be just a crystal set in which the tuned circuit is replaced by a varicap tuned circuit with the rectified output going to a display. The voltage applied to the varicap is a sawtooth waveform (for example derived from a 555 timer IC) which also drives the x axis of the scope: this makes the tuned circuit sweep a band of frequencies over a second or so with the receiver's rectified output connecting to the Y axis of a scope. The resulting scope trace is then a picture of the band being scanned.
|Shortwave scan on the VK2ZAY mint tin spectrum analyser|
An earlier simple design is available at http://www.qsl.net/n9zia/spec/Homebrew_Spectrum_Analyzer.pdf which uses the sweep voltage to control the local oscillator of an existing receiver.
More complex designs have wide dynamic range, more linear displays and narrow IF filters to improve the resolution. "You get what you pay for", but simplicity is still capable of providing something very useful.