First, watch this video (I know, the narration is slow and the music is hokey, but it does a pretty good job of explaining what’s going on when you shift)
So, you rev the engine with the clutch still in to match revs for the lower gear you’re about to put it in. Let’s say 2nd just to keep things clear.
This means that your engine (if you do it right) will be turning as fast as it will be turning when in 2nd gear with the clutch out. But! because the clutch is in when you rev, you aren’t getting the gear on the input shaft to spin at the speed it will be spinning once you’re fully in gear.
That’s where the synchro (about 3:47 in the video) comes in. It gets both sides spinning at the speed they need to spin via friction.
Old transmissions didn’t have synchros, which meant drivers had to get both sides of the transmission spinning at the proper speed to engage. By letting the clutch out in neutral and revving the engine, the input shaft spins faster to match what the other side of the transmission is already doing.
If you don’t let the clutch out in neutral, the input shaft doesn’t speed up, and the gears don’t go together nicely without the help of a synchro.
Some people who are especially anal about minimizing wear will double clutch even with a modern transmission to prevent synchro wear. Not really super necessary - those synchros will last a long time - but there’s nothing wrong with doing it either.
Incidentally, double clutching does come in handy when you have a bad synchro, as you’re doing its work for it so that it doesn’t have to.