Do battery saver apps really work? Is it possible to save battery programmatically? There are numerous apps in the Google Play store claiming to double or triple the battery life. Is it just a myth? Or can they really?

If they really do save some power, why doesn't Android do it by default?
Those battery saver apps probably work by killing tasks, lowering brightness, disabling data or internet connection, and several other things. Those little things will definitely save battery one way or another.

Why doesn't Android do it by default? I don't know; you'd have to ask the Android developers that. If Android did what battery saver apps do, they'd be limiting Android. Who wants their screens to always be in low brightness? Or their internet connection off when not in use? Those things are probably better off for people to decide, and battery apps (and other relatated apps or utilities) will help them do that.

See also this post on improving battery life: What can I do to increase battery life on my Android device?