The transfer windows are essentially the "when you can register a new player" window. You can sign someone outside of a window but you cannot register them with the FA/Premier League until the window opens. Adrien Silva for Leicester is the best example of this... The transfer went through seconds after the window closed and that meant Leicester were unable to register him until the next window.
This new player may also be under the registration age for Premier League games, but he would need to also be 'homegrown' to qualify under the rules. As he wouldn't have spent at least 3 years at a team in England (Welsh teams competing in the pyramid also count towards this) between the ages of 16 and 21, he wouldn't be allowed to play in the Premier League.
I hope I explained that clearly