The Space Cadet is a keyboard designed for use with Symbolics' Lisp Machines, and many of the extra keys are specific to that use.
Starting from the bottom row:
- Hyper, Super, Meta and Ctrl are modifier keys, known as bucky keys; they are equivalent in use to the Ctrl, Alt and Alt Gr on modern keyboards (and to the Windows, context menu, Apple and Option keys). They allow shortcuts to be built using the main keys. You can see their heritage in Emacs, with its many shortcuts involving Ctrl and Meta (Alt or Esc on modern keyboards). Many Linux systems use the Windows key as Super.
- Top, Greek (or Front on some models) and Shift allow the various symbols on the keycaps to be produced. Thus M on its own produces "m", ShiftM produces "M", TopM produces "≥", GreekM produces "µ". Repeat allows keys to be repeated — they didn't repeat by default.
- I'm not sure what Mode Lock or Line do. Alt Mode was used on ITS and was considered obsolete even when the Space Cadet was designed. Rub Out is the ancestor of our Backspace. Help was supposed to be handled by applications for on-line help (but rarely was).
- I, II, III and IV were used for quick selection in menus.
- Network, Terminal and System sent escapes to specify which part of the system (as a whole) the user wanted to interact with.
- Status was supposed to ask the system to provide status information for the currently process.
- The thumb and finger keys are the ancestors of our arrow keys, but they weren't used much.
- Macro was intended to provide a keyboard-based macro system, but it saw little use in practice.
- Quote was supposed to allow escaping other keys, so they could be sent — a bit like the Esc key on Atari 8-bit keyboards.
- I'm not sure what Over Strike does.
- The two Clear keys clear the current input field and the whole screen respectively.
- Hold Output and Stop Output provide output flow control (similar to CtrlQ and CtrlS).
- Abort, Break, Resume and Call provide process and function control.
With the Greek key, these keyboards allow all of the APL character set to be entered easily. The modifier keys can be combined: GreekL produces "λ" as you'd expect, and ShiftGreekL produces "Λ". Likewise the bucky keys can be combined, so CtrlAltA is meaningful — this is still the case on modern keyboards, the obvious example being CtrlAltDelete. The Greek-style behaviour lives on too; on my Linux systems with a French keyboard layout, Alt GrL produces "ŀ", and ShiftAlt GrL produces "Ŀ". On the Space Cadet, the modifier keys were specifically laid out so they could be chorded single-handedly using either hand: you could press Ctrl and Alt with your left hand, and L with your right, or vice-versa e.g. for Q.
You'll find more information on Mike McMahon's page on the Space Cadet and in John Kulp's design notes for the keyboard. Thanks to Mark and mnem for further information provided in the comments!