What is a UTX 0 Proof? (as discussed in this video) This video discusses how off chain transactions can be accomplished as an alternative to increasing the block size.

At the 21:25 marker he discusses "UTX 0 Proofs".  What is that?
UTXO is short for "Unspent Transaction Output". You can imagine the UTXO set as a set of tuples of the form (txid, index). Transactions remove items from that list by claiming the outputs and add new items by allocating the bitcoins from claimed outputs to new outputs. Basically the whole of Bitcoin boils down to agreeing on what the current UTXO looks like, as it tells you how many bitcoins are owned by each address at any time.

The UTXO Proof Peter was talking about is a way for a node, that does not store an entire copy of the UTXO set, namely a DHT node, to validate transactions. This is done by querying other nodes in the DHT and retrieving the parts of the UTXO that is needed in order to validate the transaction.

Notice that, while there are some ideas on how to implement this, there currently is no prototype.