"Wait for Move's Completion" means that the selected route needs to be completed before starting next step.
Example if the event says following:
@>Set Move Route: Player
: : $>Move Down
: : $>Move Down
@>Wait for Move's Completion
@>Text: Hello!
@>
These lines is saying to the game:
Move the Player Down Two steps, and then execute the text message: Hello!
If the player is able to go down one step, but not the other, then the message won't appear and the events goes into a kind of lock, where you won't be able to do anything before the subject blocking the way is removed.
If "Wait for Move's Completion" dosen't exist the player will start to move at the same time as the textbox appears.
If "Wait for Move's Completion" do exist the player will move and not until the whole route is completed the textbox appears.