Quick explanation for "home-grown" for those who don't know....
For a player to be considered home grown (trained at club), they have to be trained at the club for 3 years between the ages of 15 and 21. For example, Cesc Fabregas is considered a home-grown (trained at club) player for Arsenal, as he first signed for them when he was 16.
To be classed as home-grown (trained in nation), it's similar, except that he has been signed after he gets home-grown (club) status with another club in the same country.
Just look at the Man Utd squad ....
Amos, Fabio, Rafael, Evans, Scholes, Fletcher, Giggs, Cleverly and Welbeck are all home-grown (club) players with Jones, Ferdinand, Smalling, Carrick, Young, Owen and Rooney are all home-grown (nation) players. That's 16 players, which is more than enough for the FA* and Uefa** squad limits for competitions. (I've probably missed a few players off that list, but I've not included players that only get the odd game.)
* Squad limit of 25 players, of which 8 (IIRC) must be home grown. Players under the age of 21 do not have to be registered, but can still be used.
** Squad limit of 25 player, of which 8 must be home-grown (nation). Off those 8 players, 4 must be home-grown (club). Players under the age of 21 do not have to be registered, but can only be used if they have spent a minimum of 2 years at the club.