Before anything else you need to configure everything
Enter the address you're interested in the address box. If you've copied an address to the clipboard you can use the paste button or menu option to paste it into the address box.
Hitting return or clicking the Go button (the purple arrow) will identify the owners of the address using whois by default (it can be customised via the startup script
You can do all sorts of things to that address using the toolbar
The results will appear in a new window. Some parts of the results may be highlighted as active text. Right-clicking on the active text will popup a menu of things you can do next. Left clicking on it will do the most common operation.
If you're chasing down information about the owner of an address it's handy to be able to save the useful intermediate results. The two buttons on the left of the toolbar let you copy results to the Log Window for later use
If you have a suspicious looking set of email headers you can copy them to the clipboard from your mail reader and paste them into Sam Spade. They'll appear in a new window, with any addresses highlighted as active text. If you have Smart Paste enabled Sam Spade will run a rudimentary header analysis. It's not perfect, but can sometimes find suspicious parts of the header
Header analysis is only useful if you paste a full set of headers. See here for ways to turn on full headers on common mail programs
There are some more sophisticated tools available on the Tools menu. Some of these may need to be enabled before you can use them.