Why the Principle of Relativity is mentioned so much?
The Principle of Relativity is the basis of Einistein's reasoning. Without it, the Special Theory of Relativity would not be there.
Despite its misleading name, the Principle of Relativity is really about objectivity. Maybe a better name is Equivalence Principle, as used in the General Theory of Relativity.
Basically, it means that a physical law discovered in one lab can be verified by another lab, even that lab is on a train or a plane. A law found here on Earth can be used on the Moon. A conclusion based on the facts from the Milky Way can be applied to other galaxies. If any contradiction is found, the rules we originally found out may be wrong, or at least incomplete.
In other words, this principle says that similar causes create similar effects. No matter what viewpoint you take, the final conclusion should always be the same.
For me, the easiest demonstration of this principle is through Newton's second law of motion. If you can found any easier one, please let me know.