Because you're both connected to a common lightning node, at least indirectly. There are several public lightning nodes that accept connections and they are joined together. Connect to one and you have access to their network.
> Similarly, if I want some random person to send me money, how do I convince them to run a lightning node and to let me connect to it?
Same as above. The lightning network is, of course, only going to work with other nodes who are also connected, but there is an incentive to join, since when you do, you can transact with all of its peers.