Stateful Multilayer Inspection firewalls monitor the state of active connections and determine which network packets to allow through the firewall. They are designed to inspect the TCP handshake, which is the initial connection setup process between two hosts in a network. By monitoring this handshake, the firewall can determine whether a requested session is legitimate. This technology allows the firewall to not only filter packets based on predefined rules but also to ensure that the packets are part of an established and approved connection.