The path difference refers to the paths followed by the waves. That's where the d sin theta bit comes from. The point is that light enters the guide over its whole width. When it comes out, it is essential that its
actual path (not the horizontal distance) inside is the same for all the light - or an integral number of wavelengths different. So, if the route via an extra double bounce is n lambda different from the route involving no extra bounces then this condition is met. If it's not n lambda different then you can get cancellation of some rays by others.
Look, you can't have a difference unless you are comparing two paths (not reflections) - isn't that obvious? It's what happens on the actual way out of the guide. Some rays will come out with two more bounces than others and will be mis-timed but the mis timing doesn't matter if its a whole number of cycles. Two of those diagonal lines represent this extra distance traveled by those rays on the extra two bounces.
Have you seen the Newton's rings and 'oil film' stuff in textbooks? It's the same thing at work; two different routes through the system need to be n lambda different for a maximum of constructive interference.
I have told you more than enough by now for you to work out what it's all about, I'm sure. Go back and look at other examples and explanations of interference and
diffraction if it still isn't clear and start from scratch.
