When opening a LN channel using LND, what does 'push-amt' do? I set up Lightning node (LND) and wanted to receive a payment. Autopilot for managing the channels always creates channels w/ zero remote balance, thus nobody can route a payment to/through my node. 

The command for manual channel opening is lncli openchannel [command options] node-key local-amt push-amt. If I specify nonzero push-amt, the remote balance is then, unsurprisingly, nonzero. 

But what I find very confusing is "whose" money that was? Have I just given my peer free bitcoin?

The help command for openchannel push-amt states: the number of satoshis to push to the remote side as part of the initial commitment state (default: 0).
The money now belongs to the counterparty in the channel you opened. In every one of your channels, whether you opened them or they were opened to you, "local balance" refers to your spendable/forwardable balance and "remote balance" refers to the counterparty's.

Don't worry though, if your node stays online reliably, then others will open channels with you, giving you inbound capacity. At this stage there's not much else you can do.