Distributed systems according to Lamport?
My interpretation of this quote is that if we consider the whole distributed system as a black box containing multiple local processors, then when we send input requests to this box, the whole box moves to another state even though we don't know what that next state is (particularly in non-deterministic state machines), but we are sure that there's no inconsistent interpretation within the box.

Actually input requests could be thought of as an external node that does atomic broadcast to all internal nodes (within the box). This type of broadcast ensures us that all internal nodes receive messages in the same order, so all local state machines move to their next states. If we define state of the whole system as a set of all internal local states, so whenever we receive an atomic broadcast (i.e. input request), our system's state moves to a next state depending on the order of messages that all internal nodes agree on. Hence, each different order leads us to a non-deterministic destination (the next state) in the state machine of the whole system.

Therefore, total ordering is sufficient to model a distributed state machine. However in some applications some more properties might be needed, for instance, FIFO ordering, causal ordering, etc.