One of the earliest examples of proof of burn that I can remember was around 2013 where counterparty was selling the first (?) tokens using the bitcoin network. Their application used an invalid multi-sig address to send your bitcoin to that would effectively burn it as there would never be a private key that could spend it (can't remember the full details here). Addresses that sent money in would then have tokens available to use on their application layer.

I believe this is actually where the term was first coined as well. What it means now I'm not sure, don't follow crypto.