'chutes generally come in two forms; thos epacked into rigid 'cans' which are then slotted into the a/c and those in soft packs.
In both instances the mechanism is pretty simple. The can is generally held in by fasteners whereas the soft pack is held in by a shackle.
I worked on F4 'chutes where the main chute was packed into a soft pack. On top of this goes a small auxillary 'chute which is basically a large helical spring sewn into a small 'chute.
This is held closed with a temporary pin.
The other end of the pack has the 'chutes main strop with a shackle at the end.
This shackle is pushed into the spring loaded hook in the back end of the jet and the pack pushed in after it. The 'chute bay door is then locked shut and the auxillary 'temp pin withdrawn. The 'chute is then basically primed.
The chute is activated by a handle in the cockpit. (obviously I guess)
As Phizzer says, a burst of power helped release the 'chute after use. Sometimes this would be applied when the chute was pointing at the recovery crew for extra laughs. How we used to giggle as that metal shackle sailed towards us!
Most 'chutes (Brit ones) are similar in principle. They may be bigger or smaller with cables or strops but essentially a small chute drags out the big one. Simples! (surprisingly physical and dirty work packing them though)