At block N someone could start to privately mine (empty) blocks.
They keep mining in private until block N+x is public, at which time the private (51%) chain is length N+x+1.
They then announce their longer chain.
By the protocol, this longer chain (technically "most work" chain) is the more trusted one, and undoes any transactions in N+1 through N+x.
A more sophisticated attack would include all the legitimate transactions on the network except for their own transaction(s) which they're trying to double spend. That way the network isn't disrupted apart from the parties you're double spending against.
You can't do that with 25% (or even 40%) hashrate.
But your chain has every block solved by you, giving you all the block rewards.
That's the magic of the 51% attack. You gain control of the blocks. Because that extra 1% isn't a HUGE margin, it may take a while for your chain to become the winning chain, but theoretically, it will happen.
It's true that you can't synthesise false transactions, but you can undo anyone's transactions, not just your own.