Quick mode can be activated by pressing Q while in a replay (or by pressing the button you found). It should then do some initial work on the replay and finally refresh the world.
Once activated, only block (not tile entity) changes, entity spawning, entity despawning, entity movement, weather and time of day will be replayed. In exchanged for the reduced quality of replay, you can almost instantaneously seek to any time in the replay, both forwards and backwards.
Please do not report lack of anything which isn't listed (e.g. particles), this is intentional and for most things will not be fixed (there have to be some tradeoffs). Quick Mode is also only available on 1.9.4+.
(And no, you've not missed it in the docs, it has yet to be documented.)