The Lorentz transformation implies that moving clocks and moving lengths are related differently from Galilean intuition.

Time dilation

If two events occur at the same place in a clock's rest frame, the clock measures the proper time $\Delta\tau$. An inertial frame in which that clock moves measures

$$\Delta t=\gamma\Delta\tau,$$

where

$$\gamma=\frac{1}{\sqrt{1-v^2/c^2}}.$$

The interval between the clock's ticks is therefore longer in the frame where the clock is moving.

Length contraction

If an object has proper length $L_0$ in its rest frame, an observer for whom it moves along its length measures

$$L=\frac{L_0}{\gamma}.$$

The endpoints must be recorded simultaneously in the measuring observer's frame, which is why length contraction is inseparable from relativity of simultaneity.

Symmetry between inertial observers

For uniform relative motion, each inertial observer can describe the other's clocks as time-dilated. There is no contradiction because statements about distant simultaneity differ between the frames.

These are coordinate relationships between measurements in different inertial frames, not mechanical slowing or compression caused by motion through an absolute background.