I don't understand a way of proving PSPACE-completness. The way was used by my lecturer. I can use reduction, however following method confuse me:  We consider sequence (of polynomial length) of configurations and check if transitions betweens configurations in sequence are legal. We also check if machine accept word $w$. In other words, we analyze run of polynomial-space machine $M$ on word $w$ and check if this run is legal and accepting. I can't understand why such reasoning work. I don't show exact example because my lecturer use this technique fairly often.  Can you explain me why it work? 
Confusing method of proving PSPACE-completness
complexity theory;complexity classes;space complexity