Miners operate nodes (as do many non-miners). These nodes make connections to each other. When a node receives a block or transaction from another node, it will check the block or transaction and then send it to the other nodes it is connected to. In this way, a block or transaction reaches every node in the network.
The miners do not connect to some central server or service in order to broadcast blocks to the network. They simply send the blocks to the nodes that they are connected to and the blocks are then relayed to all nodes.
As for how a node figures out who to connect to, please see this question: How does the bitcoin client make the initial connection to the bitcoin network?