Why it works for the Titans
Kenny Britt has been a train-wreck in the Titans controlled, short passing offense and he seems to get flagged every other play for a block in the back or a hold. Getting anything in return is a blessing. Mark Ingram isn't a prolific runner, but the Titans could put Shonn Greene on IR, and use Mark Ingram as a #2 or #3 running back.
Ingram is a perfect compliment to Chris Johnson and would probably be an upgrade over Shonn Greene both short and long term. The Human Pinball AKA Mark Ingram, may not have been as highly touted as Trent Richardson but look at the stats.
Mark Ingram: 295 career carries, 1,107 yards, 3.75 YPC, zero fumbles and 10 TDs
Trent Richardson: 311 career carries, 1,090 yards, 3.5 YPC, three fumbles and 12 TDs.
Bet you didn't expect that.
Why it works for the Saints
The deal sounds so good to the Titans, so why should the Saints do it? The Saints have a decent #1 WR and an elite TE, but they don't have any veteran help outside of that. New Orleans could use a big wide receiver to use opposite Marques Colston.
Also, what do they have to lose? So far this year the Saints have only rushed Mark Ingram 17 times and it looks like he is fourth on the depth chart behind Pierre Thomas, Darren Sproles and Khiry Robinson.
Both teams get new pieces that fit their offensive philosophy. The Titans get a "between-the-tackles" runner and the Saints get a big target who is in a contract year and needs a change of scenery.