https://github.com/lnbook/lnbook/blob/develop/03_how_ln_work... is a reasonable explanation of you're interested in the technical aspects.

My understanding is that you start by doing an actual BTC transaction of, say X USD, for which you pay the fee, and then you can "spend" up to X-fee USD in the lightning network. People who receive payment from you can opt for getting a BTC transaction from your stash or use it to do other payments.

The base idea is that it uses the BTC network as a dispute settlement fallback. "It works but is expensive, let's just exchange through the lightning network".

You can see Lightning as being a network that exchanges "BTC IOU". If the network decides for some reasons to be hostile to you and refuses to broadcast the transactions where you receive money, you still have a signed token that proves you have the BTC IOU and that are acceptable by the BTC network (for the cost of a fee)

So it's like withdrawing money from the bank as cash to use. I get that, but after I promise the money to bunch of people how does the system prevent me from double spending or just redepositing my money to my account?

Lightning is basically a way to avoid trading on the Bitcoin blockchain by passing transactions through third parties.

Its like a giant bar tab. You have this liquidity pool with the network participants do all their transactions and go back to the chain when when they decide to settle.

Essentially it nets off contra positions so they don't have to hit the chain in a gross basis, but only on a net basis.