Understanding the proof for time contraction in special relativity

In this stack about deriving Lorentz transformations, the op writes an answer where he proves the relation between time in a moving frame and regular frame but I just can’t understand what the logic of it is.

What I understand this is that when the rod moves, the path is altered and hence light needs a different time to traverse it. Noting that the speed of light must be constant in all reference frames, we set a relation with Pythagoras theorem and relate the times. So, what finally concluded was that:

$$ t’ = gamma t$$

So, where exactly do the moving frames and non moving frames come in here? And, how does this extra time which light takes to cover a longer path relate to the time contraction felt by observers in different frames?

To put it short, I’m having a hard time understanding the physical outcome of the result of this derivation