Here's my take on why A&M left the Big12 for the SEC.
1. Conference stability
2. Increased revenue
3. Increased exposure
4. Increased competition
Conference stability was key. In the late 00's, when Dodds saw how much money the B1G and the SEC were beginning to command, he realized the Big12 would never be able to compete due to it's limited TV footprint. In 2009, Dodds began working with Larry Scott and his staff to create the PAC-16 by moving 6 schools from the Big12 to the PAC-10. These negotiations and planning sessions took place without input from A&M. When rumors began to spread, A&M asked UT what was going on, and was told, "Don't worry, we'll take care of you." In 2010, UT tried to force A&M's hand to join the PAC. The other 4 Big12 schools were on board with the move but A&M tapped the brakes because the PAC was a horrible fit for A&M. A&M said it was looking at all its options so Dodds told A&M AD Bill Byrne that if A&M didn't fall in line UT would never play them again in any sport. A&M was given 72 hours to decide.
It looked like the Big12 was about to dissolve. Seeing this, Nebraska reached out to the B1G and secured a spot to start playing in that conference in 2011. Colorado, afraid its place in the PAC-16 might get poached by Baylor, signed on with the PAC a day early to ensure its spot. Colorado fully expected at least 4 of the other 5 Big12 schools to join the following day. This was critical for Colorado as the remaining Big12 schools would ensure a manageable travel schedule and preserve some Big-8 rivalries. What those two may not have known was UT had been simultaneously negotiating with ESPN and FOX to create its own network. Initially it was looking like the LHN could garner about $2 million per season for UT. However, when ESPN saw that the PAC 16 would force massive (read expensive) realignment in addition to losing big brands like UT and OU, it decided to offer UT a $300 million dollar poison pill rather than spend billions renegotiating conference TV deals en masse. UT tried to get the PAC to allow them to have their own network rather than be part of the PAC Network as originally agreed upon. The PAC balked and UT backed out of the deal. Once UT was out, the PAC scuttled the rest of it and went with its first alternate, Utah... creating the PAC-12.
Once UT showed it was willing to destroy its conference to get more money, it became clear the Big12 would eventually be unable to satisfy UT's revenue needs. UT would leave the Big12 at some point in the future... maybe to the PAC, maybe to the B1G, maybe independent. Regardless, it was clear that UT would bolt someday. A&M didn't want to be in a position where it was forced to follow UT into a situation that wasn't right for A&M. The only way to prevent this was to determine its preferred destination and work toward that goal.
A&M's desire to leave was kicked into hyper-drive by Dan Beebe's handling of the LHN. Big12 mandated that school media platforms like the LHN would only be able to televise 1 non-conference home game per year. No discussion had occurred regarding televising recruits, there was no discussion about UT paying conference members to have their game with UT moved to the LHN, and it was understood that ESPN wouldn't force Big12 schools to appear on the LHN. However, after the LHN launched, UT moved forward with all of these. When these issues were brought up to Beebe, he refused to draw the line. A&M had to appeal to the NCAA regarding the plan to televise recruits. The NCAA responded that such an activity would be a violation and recruits' games were never televised on LHN. Had it just been up to Beebe, UT would have been allowed to do it. It became apparent to A&M that Beebe intended to favor the Longhorns as much as possible to keep UT's finger off the Big12 destruct button.
So A&M reached out to the SEC in the summer of 2011 to pursue membership in its conference. The SEC was receptive. The remaining Big12 schools indicated they would sign a waiver agreeing to forego any torts that might otherwise be pursued. Things were moving smoothly. A snag was encountered when OU also became frustrated with Beebe. In an effort to oust Beebe, OU stated they were looking to move to the PAC on their own. Baylor was so alarmed by this that they contacted the SEC to tell them they were no longer going to sign the waiver. Us Aggies saw this as an attempt to keep us from going to the SEC when in reality it was a warning shot to the PAC that OU wasn't getting out without a legal battle. Baylor knew the Big12 would survive just fine without A&M but losing OU would be a major blow to the league and might result in its dissolution. In the end Beebe was ousted and Baylor signed the waiver, paving the way for A&M to join the SEC.
Once the Big12 had been destabilized, A&M no longer had faith that the conference would be viable home long-term. With the clock ticking, A&M determined its own future rather than letting fate decide it for them. Had the 2010 Big12 Missile Crisis never happened, A&M would still be in the Big12 today. But once that Rubicon was crossed, the conference has never been the same.
In the end, A&M's belief that UT would leave the conference for more money was validated. The surprise was that it came before the LHN contract ran out and that the SEC was the chosen conference.