Based on the two previous answers, this is my attempt from a microscopical point of view:
The gas near the valve acts as a "wall" against which the air inside the tyre is expanding. When particles inside the tyre "push" this "wall", they lose Kinetic Energy and they cool down. Adiabatic expansion
The energy transferred to the particles that form the "wall" near the valve gets lost when these air molecules exit the valve. Joule effect
Thus the Kinetic Energy of the particles inside the tyre gets lower and lower --> lower temperature.
(Example 6.10 from Fundamentals of Engineering Theormodynamics by Moran)