The blockchain knows UTXOs and locks (locking scripts). An address ~= a public key.
When you sign a transaction with your hww you give your wallet a PSBT which contains the UTXOs you want to spend, and the UTXOs you want to create.
Something which does have a view of the network is what creates the PSBT, so it knows what UTXOs are yours.
Both the hww and the thing that makes the PSBT knows your public key (xpub) and using that info can determine 1. What UTXOs are yours, and 2. That it can sign for the UTXOs
You prove that you own it by signing for it
Hopefully that helps