How does proof of work ensure network security?

I have been looking into how blockchains work and I can't seem to find the technical reason as to why proof of work actual ensures that all transactions are valid.

From what I understand, individual transactions are verified through asymmetric keys, ensuring that someone can't spend crypto that they can't prove they own by using their private key.

So what attacks does the proof of work system prevent? From my research I find that it helps prevent double spending attacks, but I don't understand how it does this, technically, given that anybody can mine a new block given a decent amount of resources, even without controlling 51% of the network. What am I missing? How does PoW protect the network?

Edit: Got it now! I was confused because I didn't understand the double spend attack. I was seeing the double spending problem as a sender respending the same currency that they spent previously in the chain, when the actual problem would be the sender spending the amount, waiting for the chain to verify the transaction, and then creating a new fork of the chain starting at the block before they spent the currency. With proof of work this is computationally impractical because by the time the network confirms the first transaction, there is no way (statistically extremely difficult) that the malicious sender could compute the proof of work for every block in the modified blockchain fork and force it to overwrite the previous chain. The white paper answered my questions. I didn't realize that it was only 8 pages. Thanks for the help!