As I noted in the comments, I'm somewhat blown away by this definition, but I'll take a stab at it. You've got a set, and the points in it are moving around with the passage of time, and $\Phi$ is keeping track of the movement. Think of $T$ as giving the time. $\Phi(0,x)=x$ says that at time zero the points are at their starting places. The third condition says that if you look at where the points are after $t_1$ seconds (or days, or centuries, whatever), and then you look at where they are $t_2$ seconds later, you find out where they are after $t_1+t_2$ seconds.
Only everything's being done in greater generality. I've been thinking of $T$ as time, which is naturally an interval in the continuous case, or the integers or naturals in the discrete case, but whoever wrote that definition wants to allow $T$ to be an arbitrary monoid. And I've been thinking of $M$ as a set of points on the line, or in the plane, or 3-space, but the author intends for it to be any abstract set. So I haven't so much explained the definition, as given you one very specialized (but, I think, very important) setting it which you can interpret it.