How to understand pushdown automata intuitively?
Intuitively, a pushdown automaton uses a stack and using a stack we can do a depth first traversal of a parse tree. It means that we can accept strings which are in context free languages by using stack of a pushdown automaton (left-most derivation). This is not a rigorous proof that languages of pushdown automata are context free. For proofs you must see various textbooks on the subject.