Look at this fromthe view point of the person in frame s. At t= 0 let both frame s and s’ have their origin of coordinates conicide and let their x and y axes be parallel, repsectivelt yo each other. Let the light clock in the s’ frame be aligned along the y’ axis. A person in the s’ frame then sees light bouncing back and forth along the y’ axis.

Now let s’ move in the +x direction. An observer in s sees a light clock in s’ move so the light travels both in the x and y (note no primes – the s frame observer has no knowledge of the s’ coordinates). But he also sees the light take a longer path to make one round trip.

Let’s say the clock is set up so that in s’ one round trip = 10 nanoseconds – the clock is 1.5 m long. But the person in s sees the light trave teh length of the clock and some distance in x as the clock moves. Since the y and y’ directions are perpendicular to the motion, both observers agree that light travels a distance of cT in T seconds in y. But the s observer sees the light in th eclock travel a total distance of

d = cT*sqrt(1 + (v/c)^2) where v is speed that s’ is moving relative to s.

So the observer in s will say that one “tick” of s’ clock is

t = d/c = T*sqrt(1+(v/c)^2) > T

Hence the observer in s concludes that the clock in s’ is running slow – t > T. The speed of light doesn’t change.