403b/401k roll over to your new employer when you leave. The amount of money that you put in from your paycheck is always yours (plus or minus market change but don't get hung up on that especially early on). The employer match takes a certain amount of time to become yours. This is called vesting. At my current employer, you vest 33% every year so after 3 years of working 100% of the employer contribution belongs to me. If I leave before then I get a prorated percentage.
Using fake numbers: I contribute $100. My employer matches $100. I have $200 in my retirement account. If I leave after a year, I keep my $100 and $33 from the employer. If I leave after two years, I keep my $100 and $66 from the employer. If I stay until I'm fully vested, or three years in this example, I keep $100 I put in plus $100 employer puts in.
So even if I leave before fully vesting, I have money from they employer that I wouldn't have otherwise had. It's the best "return" you'll get on your money. And when you contribute more younger, compound interest has much longer to work it's magic and your money grows to much more.
Google retirement calculators and look at the difference in contributing a set amount starting now versus in 10 and 15 years. Starting young means you don't have to play a ton of catch up later.
This is a good quick intro with visual aids that I think explains it more clearly than I do:
I’m only 22. Should I contribute to my company’s 401k? | NerdWallet Investing