Why is there a maximum landing weight limit?
At the instant of touchdown, any vertical kinetic energy the plane possesses must be absorbed by the landing gear, which thereafter must bear the entire weight of the plane. That kinetic energy depends on the sink rate and the plane's weight. 

The sink rate ideally will be zero at wheels down in a perfect world, but the gear is designed to accomodate a certain range of sink rates and plane weights via compression of its spring struts and shocks. If either the sink rate or the plane's weight causes the maximum compression range to be exceeded, the suspension will run out of travel and bottom out hard and the resulting shock can damage the attachment points of the gear to the fuselage or the gear itself- or even cause the gear to collapse. 

The maximum gross weight at takeoff (full tanks, maximum payload) is set by engine power and wing lift when configured for takeoff and the expectation is that by the time the plane has reached its destination it will have burned off enough fuel to be under the allowable gross weight at landing (which is set by the design of the landing gear as explained above). This means that the plane's designers can minimize the weight of the landing gear by not having to strengthen it enough to routinely accomodate a landing at maximum gross weight at takeoff. 

And that, in turn, means that if a fully-loaded plane has to make a quick return to the line for any reason, it must dump fuel first.