"I was thinking the other day about how, as I understand it, Fannie and Freddie were created to allow lenders lend more money. The customer-facing lenders make loans for houses, and then Fannie and Freddie use tax dollars to buy those loans, so the retail lenders can lend more money."

This is exactly what they do, as a matter of fact, except for the source of Fannie/Freddie money. When they buy up mortgages, they securitize (package) them for investment from other entities. This investment capital is then used for other operations, including putting more money back on the market (they also raise money via investment in overnight commercial paper, asset backed securities, getting their spreads from the euro/fed funds markets, and so on). Freddie/Fannie also sometimes get seller/servicer (administrative) fees if they're the ones dealing with the payment/escrow for loans/a batch of loans.