Lightning network invoices reveal the destination of payments. You might want to proxy an invoice's destination because you need help finding a path to the original destination, because the original destination is your public lightning network node and you want to keep that private, or because you are paying from a custodial service and don't want the custodian to know the destination of your payment. In order not to have to trust an intermediary with your funds, you can request that an lnproxy relay generate a "proxy invoice": an invoice with the same payment hash.
Then, once you verify the amount and payment hash, you can use the proxy invoice wherever would have used the original invoice and know that the only way for the payment to succeed is for the lnproxy relay to pay the original invoice.
Proxy invoices are hodl invoices. When an lnproxy relay accepts an htlc for the proxy invoice, it immediately pays the original invoice and uses the revealed preimage to settle the proxy invoice. This ensures that you don't need to trust lnproxy relays with your payments.
For additional privacy, using a vpn or tor prevents the lnproxy relay from discovering your IP address. Onion routing on the lightning network hides the source of payments to proxy invoices.
The source code for running an lnproxy relay is free and it's easy to use lnproxy.org with any relay server (just paste the url into the "relay" field).