Why does chemisorption increase with rise in temperature, even though it is an exothermic process?
To answer that first think about a reaction like combustion reaction. Even though combustion reactions are exothermic you need to provide heat to the system by increasing its temperature. See, every reaction or chemical change happening requires a certain minimum amount energy called the "activation energy" for the reaction to start. This energy is required to break the bonds of the reactants so that they can react to form product. That's why we say that at low temperatures, kinetics controls the reaction rather than thermodynamics. Now, once you are at a temperature which is able to provide the sufficient activation energy to the reactants, the thermodynamics comes into play which essentially says in this case that if a reaction is exothermic, increasing the temperature will shift the reaction in backward direction. 

Now coming to chemisorption, in this case too you need to increase the temperature to provide sufficient activation energy so that the reaction can start. So increasing the temperature increases the rate of reaction. After a certain temperature when thermodynamics become more important, the reaction shifts backward.

Hope it helps!