You may have noticed that a channel is often visually represented as a bar with two colors to represent how many sats you can receive and how many you can send. The act of receiving a payment is the movement of sats from the receive side to the send side.

The sending lightning node will calculate a route over the lightning network to your receiving node. A route consists of hops from one channel to the next until it terminates at one of your 15 channels. So, where the sats "go" is one of your 15 channels. The route will ensure that there are enough sats on your channel's opposite side, so the final step to receiving a payment is moving the sats from your peer's side of the channel to your side.