The developer options menu gives you the ability to tweak a few "hidden" settings and enable USB debugging. That last gives you more options to modify software from a computer using USB. The other settings may depend a bit on the phone model, but let you do things like adjust the system animations, adjust what is written to system logs, show detailed process statistics (i.e. the sort of thing that a developer might have a use for but most people don't) and make some adjustments to gpu use and memory usage which are best left alone if you don't know what you are doing.
Some of these things might have minor effects on perceived performance (e.g. system animation settings), but no, enabling developer options will not magically improve the phone. And some of the options could degrade performance badly if you don't know what you are doing (e.g. telling it not to keep processes could sound attractive to someone who has been misled about how android manages RAM, but would be a PITA in practice and would more likely reduce battery life). So by all means activate the menu, but don't mess with stuff without finding out what it is first, and take a note of the original settings before changing things.
