Question

Why do the rate of reaction for a reversible reaction slows down at equilibrium stage.

Answer 1

At equilibrium, the forward and reverse reactions are still happening, but they are equal in both directions. The NET reaction rate (forward - backward) has to be zero at equilibrium.

Answer 2

Everything DrWLS wrote is true. To add to this, the RATE of the reaction slows, as one nears the point of equilibrium, is because the concentrations of the reactants is decreasing. For example, in
A + B ==> C + D, initially (A) and (B) are high, there are more molecule of A and B than anything else, the change of collision between A and B is high so the reaction is fast. As time passes, the concn A and concn B becomes less, fewer molecules mean less frequent collisions and the reaction slows.