The reasons why Rutgers ended up in the Big Ten are many.
Here is the real reasons, like em or not, facts are facts:
Penn State wanted the Big Ten to add schools that they could just drive to for years. They wanted Rutgers, Pitt and Cuse, all three or at least one of those three. Cuse dropped out of the AAU and out of consideration. Pitt didn't offer anything to the BTN that PSU already did, so the choice was clear.
BTN wanted Rutgers since New Jersey is part of the NYC TV Demo, the largest in the US.
OSU and Michigan pushed for Rutgers once they saw the numbers.
Many of the Big Ten schools wanted more exposure in the Northeast where many of their alumni are located, playing in NJ does that.
Rutgers is an AAU member and after the merger with UMDNJ and top 25 research university which made the Big Ten Presidents who are the ones who sign off on new members very happy.
The Big Ten was talking to Rutgers (keeping them in the Loop) since at least 2010.