If a macroscopic body of mass $m$ moves according to a certain law of motion like, for example, $$x(t)=A\cos(2\pi ft)$$ then what physical interpretation can be attributed to the Fourier transform of $x(t)$?
It tells you what is the spectral content of the motion.
Some examples of when this might be interesting:
The object being measured is a point on a guitar string---then it would tell you what note is being played
The object being measured is a planet. Then the peak frequency of the motion is the inverse of the planet's year.
The object being measured is part of a mechanical actuator---then the spectral response due to an applied force impulse (or driving signal) could tell you about the requirements for a control system for that actuator to remain stable. Generally in a linear system if you can measure the spectrum of the response to a know input with known frequency content then you can predict the response to other inputs.