Throughout the campaign you may find yourself either affiliated or presented with the opportunity to affiliate with the various organizations in Boston.
There are many ways this can happen, for instance debt or blackmail. Ideally it will happen when you suitably impress their representatives.
If you successfully attract their attention they will contact you discretely and give you an offer of "future work"
Typically you want to keep your affiliations a closely guarded secret. Letting people know who you "really" work for can be dangerous.
Once you are affilated with a group, you may receive a "strong suggestion" that you work a particular mission. You may receive various side tasks during jobs as well.
Depending on how much of a relationship you have with them, you may be expected to report to them which jobs you are working on. The responsibilities may vary, but usually they won't ask for anything that will hurt your rep.
As for why you would work for them, when you work towards their interests you will find yourself rewarded with various goodies. Depending on the job it could be anything from a safehouse to a car to high end tech not available otherwise.