Confusing method of proving PSPACE-completness
This technique is also known as computation history method. Given a Turing machine $M$ you (or another TM $M'$) inspect its computation "flow" or "history" configuration by configuration. You can think of TM configuration as snapshot of the tape content plus machine's state at some particular time. Every step of a TM generates a new configuration. The set of all that configurations is a computation history. If computation halts then that set is finite.

UPDATE: (on David Richerby comment) The following paragraph may be irrelevant to the question since I overlooked "sequence (of polynomial length) of configurations" in OP which is different from "the sequence of configurations where each is of polynomial length".

Polynomial length requirement:  because you want to prove PSPACE membership, maximum number of used tape cells must be polynomial. That's why an accent made on polynomial length. 

Transitions betweens configurations in sequence are legal: you want be sure that machine's transitions between states were really made according to the machines's transition rules, otherwise the run is not legal. For example if a configuration $C_i$ differs from the configuration $C_j$ by 3 symbols (assuming a single tape TM) then you can say that transition is not legal, since the TM cannot change 3 symbols in a single step.   

Also note that if one TM $M$ simulates another TM $M'$ then if $M'$ uses polynomial space then the whole simulation also uses polynomial space since simulation itself takes polynomial space. 

Example of use of configurations:
Assume you know that a certain class of TM machines use no more than, say, $n$ cells of tape where $n$ is input size. Then halting problem for this class of TMs is decidable. Since a TM has finite number of states $|Q|$ and alphabets $|\Sigma|$, and since we that it uses at most $n$ cells, we simply can compute that the TM has at most $n|Q||\Sigma|^n$ configurations. Then you simulate that TM and check if at some point it repeats it's configuration then there must be an infinite loop and so it never halts (since the tape use is bounded!, the TM may not move infinitely move its head to the right and right symbols). The latter class is also known as linear bound automata.