As a realist I respect what is often called “common sense.” This means that our faculties of discerning reality in everyday life are basically correct. Yes, we make some mistakes, we can get fooled by a illusionist, but we almost always agree what it is that happens when things happen to us or in front of us. These are the faculties we use to determine if a mistake has been made. Our human faculties show us reality.
The corolary to this in the sciences is twofold: (1) observation is a reliable method to gather information about the universe, and (2) the observer who is closest to an object or event is the one who is in the best position to describe the object or event. This could be called the View from Everywhere (as opposed to The View from Nowhere).
One thing this means is that an observer who is at rest with respect to some object or phenomena is in the best position to measure it properly. So the proper time, length, mass, etc. of something is the measurement at rest. What other observers measure may be different but should be related to these rest measurements. That is the origin of transformations that are made between observers, or frames of reference.
At this point there are two ways to proceed: (a) either the laws of physics are defined with respect to observers at rest, or (b) the laws of physics are independent of any particular observer. Option (b) has been chosen because option (a) does not consider how observers should relate their observations to one another. So while we might say that the real time, length, mass, etc. of something is what the observer at rest measures, nevertheless it is best to transform all observations into ones that are invariant with respect to each observer.
The result is to affirm the relativity principle (RP) that Albert Einstein articulated: “all inertial [reference] frames are totally equivalent for the performance of all physical experiments.” But it also affirms that, for example, the rest mass is the real mass of a particle (not to be confused with what is called the relativistic mass).