What tells the system which adress to send the block-reward to?
Miners include the coinbase transaction while building the block.
That is, the freshly mined block that is being relayed to the network does not only contain user transactions being confirmed (ie. transactions that you and me are sending when moving coins), but also a transaction where the miner sends themselves the block reward (ie. the coinbase transaction mentioned above).
To reiterate: The coinbase transaction containing the block reward is more or less a transaction like any other. So it's just a transaction where someone sends coins to themselves. Difference being, that this transaction may contain coins that did not exist before.