It strikes me as odd that teams who have stadiums built for their games don't use the fields for training. Is this done to make room for other sports teams to use the fields during the off-season?
The answer varies significantly by sport, and in part by team.
Those reasons that are common:
Training camp is often much more focused practice and learning time than other times (during the regular season). Think of it sort of like a retreat: you live in a more confined area, with just your teammates, and think "your sport" 24/7. It allows teammates to build chemistry and focus on their studies.
Stadiums and arenas are expensive to use - even just to turn on the A/C at something like the United Center is quite expensive. They also aren't built for training: for example, for a basketball court they've only one set of backboards. You would rather have a purpose-built facility for training: one with many courts/rinks/etc., more room for strength equipment, that sort of thing. In fact, most teams in sports like basketball have a separate training facility even during the year, sometimes in the home city, sometimes nearby. Further away is often cheaper, and quite often also closer to where the players live. (For example, until this year, the Chicago Bulls practiced and trained in the Berto Center, in suburban Deerfield; this year, they built a new training center across from the United Center in Chicago.)
For grass fields, you want to give the grass as much time to grow as possible. Each game wears on the field, so you'd rather have some games elsewhere to let the groundskeepers have more time to do their magic. You also often don't have as many fans willing to come to games in the preseason - so you hold games somewhere that people might come as a vacation, or where the people in that area might not normally have a chance to see games. Hence Spring Training for the baseball teams in Florida and Arizona - destination vacation spots that time of year - and until the last two decades, without major league teams nearby; but also with many retirees who love to see the teams they cheered for in their youth.