$N_t$ is a Poisson random variable with mean $\lambda t$ . More generally, $N_{t+s}$ - $N_s$ is a Poisson random variable with mean $\lambda t$ , independent of anything that has occurred before time s .
Define $p_j(t)$ = $P(N_t = j)$, the probability that there have been exactly j events by time t is $\dfrac{e^{-\lambda t}{\lambda t}^j}{j!}$.
Since the Poisson process $N_t$ changes only by unit upward jumps, its sample paths are fully characterised by the times at which the jumps take place. Denote by $T_0 ,T_1,T_2,...$the successive inter-event times
Apparently $T_0,T_1,T_2,...$ are a sequence of independent exponential random variables with parameter $\lambda$
Proof:
\begin{align} P[T_1>t\mid T_0=s] & = P[N_{t+s} =1\mid T_0=s] \\[10pt] & = P[N_{t+s}-N_s=0\mid T_0=s]\\[10pt] & = P[N_{t+s}-N_s=0] =p_0(t) = \exp(-\lambda t) \end{align}
Can someone please explain how the 2nd and 3rd lines come about in as simple a way as possible?