How does a PDDL solver find a solution for a given problem? As far as I know, in PDDL, an environment is designed as well as the initial state described. When we describe the target state, the solver creates some sort of a graph. How is the graph built and what are the keys (keywords) in PDDL referring to?
I know that there are many flavours of PDDL, but let's go with the standard or the most common version of PDDL.
The question doesn't really make sense: PDDL is a description language that is used to formulate a problem. This description then is the input to a planner; how the planner arrives at the intended solution is not related to the PDDL description.

There are a number of planning algorithms, and you can implement any of them to make use of a PDDL description. The output of the solver is a plan, which is usually an ordered sequence of actions, and a tree or graph structure might be a good way of capturing this.