A graphics driver update can raise frame rates in a specific game by a noticeable margin while leaving everything else unchanged. That selectivity is the clue to what a driver actually does.
The driver translates between game and hardware
A game issues drawing instructions through a graphics interface rather than to the chip directly. The driver converts those instructions into commands the hardware executes.
That translation is not fixed. There are many valid ways to schedule the same work, and some are considerably faster on a given architecture.
The driver therefore contains a large amount of decision-making about how to arrange work, and those decisions can be revised without the game changing at all.
Per-game profiles are why gains are narrow
Driver packages include profiles keyed to individual executables. When a known game launches, specific settings and code paths are applied to it.
A profile might change how memory is allocated, how shaders are compiled, or how work is distributed. These choices suit one engine and can harm another.
This is why release notes list named games. The optimisation is not general because the problem being solved was not general.
Fixes often correct a mismatch rather than add speed
Many performance problems come from a game using the interface in an unusual way that the driver handles poorly. Neither side is strictly wrong.
The driver team can add a path for that pattern, recovering performance that was always theoretically available. The gain looks dramatic because it removes a stall rather than improving throughput.
The same fault can equally be corrected in a game patch. Which side fixes it is often a matter of whichever team reaches it first.
Regressions happen for the same reason
A change that helps recent titles can slow older ones, because the code path they relied on has been altered. Older games are tested less, so this is found by players.
Drivers also drop optimisation work for hardware that has left support, which does not slow the game but stops it improving.
Keeping a working driver version is a reasonable response for anyone playing mostly older titles. Newer is not uniformly better across a whole library.
Timing explains launch-day driver releases
Developers share builds with hardware vendors before release so profiles can be prepared. The driver and the game are tuned against each other in advance.
A player who installs a new game on an old driver is running without that work. The gap can be large in the first weeks and then disappears.
This is the practical reason to update before a major launch and to be less concerned afterwards. The benefit is concentrated where the tuning is newest.