Madman, focus only on the first anim with the light bouncing up and down between the mirrors. Now imagine what this would "look" like if you were passing the still stationary mirrors - yep, it would look like the second anim to you. If this were just a ping pong ball then we'd conclude that in our moving frame the ping pong ball is travelling faster than in the mirrors' frame due to the horizontal component induced by our movement. But with light, it always goes at c (this was a basic starting point, or assumption, of relativity, not a result), so if the vertical distance between those mirrors doesn't change as we move horizontally past them, then for the light to be travelling that extra distance in our frame, the bouncing events must be less frequent than in the frame of the mirrors. Does that help at all?

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If logic doesn't work, then surely it does.