I understand how to prove that the sum of $n$ independent Poisson random variables is a poisson random variable mathematically. However, I don't understand this in terms of the concept of Poisson processes.
A poisson process must have accidents occurring at a constant rate, independently and singly. I understand that in the combined process accidents will occur independently and at a rate of the sum of the $n$ different rates but I'm not sure how accidents are guaranteed to occur singly in the combined process. E.g. below, how do you know a white and black dot won't overlap in the merged process?
Also, if someone can explain how to deduce the rate of the combined process from the concept itself instead of the mathematical proof that would be great.
A Poisson Distribution gives the probability of a number of events in an interval generated by a Poisson process. The Poisson distribution is defined by the rate parameter, $\lambda$, which is the expected number of events in the interval: expected here means "on average". It's all about the average number of events in the given time interval. Now you can understand that if you sum two Poisson random variables, with "expected number of events" in a fixed interval given by $\lambda_1$ and $\lambda_2$, then for the sum of them we simply expect a number of events given by the sum of the previous two.
About your drawing, think as if you could write a vertical line at some specific point and count how many points you have on the left of your line. This number is the expected number of events occurring in that particular fixed interval of time. Now you see that it is intuitive to expect, for the sum random variable, "on average", the sum of the number of events occurred in the two processes.