Homogeneous Poisson Process of spike train
First of all you should notice the assumption that spikes are independent and that every sequence of spikes over an interval has the same probability. That's why you can express $P[t_{1}, t_{2}, ..., t_{n}]$ in terms of $P_{T}[n]$. 
Second, note that $p[t_{1}, t_{2}, ..., t_{n}]$ is a probability density. So you can express the probability as the integral of $p[t_{1}, t_{2}, ..., t_{n}] dt_{1}dt_{2}...dt_{n}$

$\Delta t / T$ is the probability that one spike occurs in that time interval. If $\Delta t$ is small enough, only one spike is within that interval, so just let $\Delta t \rightarrow dt_{i}$. Then  $p[t_{1}, t_{2}, ..., t_{n}] dt_{1}dt{2}...dt{n} = n!P_{T}[n]dt_{1}dt_{2}...dt_{n}(\frac{1}{T})^{n}$

You should see this as a joint probability.