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Time dilation was predicted by several authors at the turn of the 20th century. In June 1905, Einstein pointed that this effect concerns the nature of time itself.

In below it’s explained it with a “Light clock”.

$L’^2=\Delta t^2_S c^2=L^2+v\Delta t^2_S$ (Pythagorean theorem)

thus,$L=\Delta t_S \sqrt{c^2-v^2}$

while $L=\Delta t_M c$

we have $\Delta t_S= \frac{\Delta t_M}{\sqrt{1-v^2/c^2}}$

where $\gamma=\frac{1}{\sqrt{1-v^2/c^2}}$ is called Lorentz factor.

Reference:

- Wikipedia: Time Dilation

- Albert Einstein: On the Electrodynamics of Moving Bodies