At its simplest it is quieting the mind. A long time ago I took a transcendental meditation course (TM) (I know, I know, it dates me, there was a girl involved) they are (were?) the folks who claimed you could defy gravity if you were good enough. There was a lot of eastern mysticism but as the scientific type I was more interested in what, if any, physiological effects were to be had. Being an undergrad in a pre-med school has some perks and one of them was being able to practice meditation wearing an EEG cap.
As far as I can tell, the primary benefit of meditation is shutting down all the noise in your head. Your brain thinks a lot, about a lot of things, often all at the same time. You might be like me where you get solutions to problems that 'bubble up' from somewhere in your brain that you didn't know was working on the problem.
Anyway, if you meditate (and there aren't any big secrets here you are simply training yourself to stop thinking without going to sleep) you clear out your brain. Like doing a power cycle reboot on a computer. Flush everything all resources available for thinking. Then when new things come up you can respond more quickly because you've got cycles to spare.
My totally non-scientific theory is that normally, when you start thinking about something new, your brain has to pick something to stop thinking about so that it can free up the cycles. If you are trying to keep a lot things in your head, that is hard because you are already trying not to stop thinking about the things you need to remember. Whereas when you just come out of meditation you aren't thinking about anything so you can quickly pick up a new train of thought.
Note that this is different than sleep because when you sleep you dream, and dreaming is a way of thinking. If you're successfully meditating, you are awake, aware of your surroundings, but keeping your finger on the 'halt' button so that you don't start up new thoughts. Then you lift your finger and off you go.