Rods
Let us put a metre-rod AB on the x'-axis of K' with the A' end of the rod
at x' = 0. The B' end, obviously, will be at x' = 1. The question is: what
will be the length of the rod in the system of coordinates K (the rod moving
with a speed v relative to K)? What we need to know is where the end A and
B of the metre-rod will be in K at the time t.
The first equation of the Lorentz transformation tells us that at t = 0:
A will be at X(A) = 0*sqrt(1-x²/c²)
B will be at X(B) = 1*sqrt(1-x²/c²)
By subtraction the length of AB will be sqrt(1-x²/c²)
The length of a rigid metre-rod moving in the direction of its length with
the velocity v is sqrt(1-v²/c²) of a metre. In conclusion the
rigid rod is shorter when in motion that it is at rest and, the higher the
speed, the shorter the rod.
At v = c the length of the rod is 0. This shows that in relativity, c is the highest possible velocity that can never be reached or exceeded by a moving body.
On the opposite, if the metre-rod was at rest on the X-axis in relation to K, the length of the rod in relation to K' would be sqrt(1-v²/c²)
According to the Galilei transformation, we would not have found any contraction of the rod at whatever speed it is moving.
Clocks
Let us consider a clock fixed at the origin (x' = 0) of K' marking the seconds
and see what happens at t' = 0 and t' = 1. The first and fourth equations
of the Lorenz transformation give us:
t(0) = 0
t(1) = 1/sqrt(1-v²/c²)
By reference to K, the clock is moving with a velocity v and, from this
coordinate reference system, the time between two strokes of the clock is
not a second, but is higher (1/sqrt(1-v²/c²) and that means that
the clock goes slower while moving that it is at rest. The higher the speed,
the slower it goes with the speed of light again being the maximum speed
possible.