Short version: during the key exchange phase a symmetric cipher is chosen and a new symmetric key is generated. All communications after that point are encrypted and, due to properties of the (good) key exchange protocol, the session key is known only to that particular client and server, meaning that nobody else can read or modify the traffic.
From the high level point of view the same scheme is used in SSL/TLS and other similar protocols: two parties start with asymmetric crypto (slow), use the key exchange protocol to generate a new symmetric key (sometimes called session key) and the actual communications are encrypted using symmetric crypto (fast).
RFC 4253 and related RFC's contain the detailed explanation of the SSH protocol.