On "Random Walk: Drunk Man in a Cliff" Problem, Why $qP$ in the equation $P = p + qP^2$?

Question

There are several solution presented in this "drunk man at the edge of the cliff" problem.

1. Random Walk of a drunk man
2. https://puzzling.stackexchange.com/questions/6415/does-the-drunk-man-fall-off-the-cliff-a-random-walk-problem

One approach caught my attention, it is straightforward but I don't fully understand it. It is required to find for P where q and p are given and this is the equation:

$P = p + qP^2$

I have two questions:

1. Why is it that the probability of moving to the right is qP? I understand why the probability of moving to the left after it is (qP)P.
2. I am more a visual learner, I can easily comprehend and easily explain in drawing. Here is how I interpreted it.

I am not sure though if I interpreted it right. Thank you in advance for shading some light.

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UPDATE

Based on the comment of @cato, I come up with this figure.

I think this is the right one.