\begin{align*} F(x)& = P(X \le x)\\ & = 1 - P(X \gt x)\\ & = 1 - P(\text{first change occurs after an interval of length} x)\\ & = 1 - P(\text{no changes in the interval} [0,x])\\ & = 1 - \frac{(\lambda x)^0 e^{-\lambda x}}{0!}\\ & = 1 - e^{-\lambda x} \end{align*}