When you look at an object from a different angle, it gets wider, shorter or whatever from your point of view. The object doesn't change, but your view of it changes.
In special relativity, those geometric transformations are extended to cover what happens when you look at it with a different velocity. The more general transformations are called Lorentz Transformations. Still nothing actually different happens to the object from its own point of view, but when you try to describe it using your own coordinates, there are some odd effects which are somewhat similar to a sort of rotation between space and time. The maths is very closely related to ordinary rotations (but uses cosh and sinh instead of cos and sin, as if the angle were imaginary rather than real).
For speeds which are small compared with the speed of light, velocities add up in the usual way and time hardly changes at all, but for larger speeds, it is necessary to adjust the rules of mechanics and electromagnetism to give consistent results from all points of view at all speeds.
Experiments have confirmed that known physical laws are consistent with this simple mathematical model, and it is now accepted as being totally accurate, to the extent that the speed of light is now fixed by definition, providing a fixed relationship between time and space units.