In my notes I have the following explanation:
The probability function of the Poisson random variable is $P_X(k)={\alpha}^k \frac{e^{-\alpha}}{k!}$ A Poisson random variable with parameter $\alpha$.
X is the number of times that an event happens in a fixed period of time $T$(or space).
This distribution is related to the binomial distribution. And what's the difference between the parameters $\alpha$ and $\lambda$?
We can see this relation if we divide the interval $T$ in $n$ subintervals of length ${\Delta}t= \frac{T}{n}$. For large $n$ we get a very little ${\Delta}t$. Each event can only happens once in every subinterval $n$, with probability $p={\lambda}{\Delta}t$.
Now, the total number of events is a Binomial $(n,p)$ random variable that equals the original Poisson$(\alpha)$variable when n $\to \infty$, with $\alpha= {\lambda}T$
Why does the probability of every subinterval $n$ equal $p={\lambda}{\Delta}t$?
EDIT: And what's the difference between the parameters $\alpha$ and $\lambda$? Usually a Poisson distribution has a parameter $\lambda$. But here $\alpha$ is used instead, as the parameter of the distribution, and later, in the definiton of the probability of each subinterval $n$, ($p={\lambda}T$), $\lambda$ is used.
If I understand you correctly, it seems that according to your note, p is the rate of successes per time interval $\Delta t$ and n is the number of trials in that interval. If that is the case, then in each interval, you run $n$ trials and have a rate of success of $\alpha$, thus it should follow that the rate of success in each interval is $p=\frac{\alpha}{n}$. Substitute this back into your formula should give you an explanation