How can a public blockchain serve as a proof of ownership, and what exactly does it prove?

By publishing a hash of the object plus your identity as a timestamped commitment — proving the object existed and was linked to you at that time, i.e. evidence of ownership, not absolute proof.

To claim ownership of a digital object later, you commit to it now: hash the object together with your identity and publish that commitment on a public blockchain, which acts as a timestamping service. Later you can show the object existed at that time and was associated with you. This is evidence of ownership — it doesn't fully prove you're the rightful owner (someone might show an earlier publication), but it's strong if no one else can. A trusted third party (e.g. a patent office) could do the same, but the blockchain lets you do it decentralized and without disclosing the object itself (only its hash is public).

Go deeper:

• Trusted timestamping (Wikipedia) — the "proof of existence" idea: commit to a hash now to prove data existed at that time.