$P_0(t)$ denote the probability that no events happened in an interval of length $t.$$$P_0(t + h) = P_0(t) P_0(h)$$ This is because if there are no events in interval $[0,t+h]$ then there are no events in intervals
- $[0,t]$
- $[t, t+h]$
These two intervals are non overlapping and it is given in question that $P_0(t)$ has stationary independent increments and so their probabilities are independent.
PS: One of the axioms of Poisson distribution is that the numbers of events in two nonoverlapping regions are independent.