Definition of a monotone machine.
When we say that a monotone machine is a set $L$ of pairs of strings $(x,y)$, the first component $x$ is the part of the input tape being read in order to produce the output $y$. 

This set is c.e. (computably enumerable) because we can run computations with all (finite parts of) oracles in parallel, and watch for pairs $(x,y)$ where this happens.  

The condition that if $(x,y)\in L$, $(x',y')\in L$ and $x\preceq x'$ then $y\preceq y'$ or $y'\preceq y$ just says that we have consistency, i.e., if you see more of the input bits you can't go back and say that you now outputted something different on an earlier input after all.

Conversely, given such a c.e. set we can produce a machine that monitors the c.e. set $L$ and looks for pairs $(x,y)$ entering $L$, and checks whether $x$ is a prefix of what's on the input tape given to the machine. If so, it extends its output string to $y$.