The Bucs went with Morris Claiborne in the first round when they could have had Trent Richardson. Martin is no Richardson, but he's a solid back and a good value here in the second round.
SB Nation's Bucs blog, Bucs Nation, explains the pick:
By picking Morris Claiborne in the first round of the draft, the Bucs addressed one need -- but they still have two major needs left: linebacker and running back. Tampa Bay has a large number of linebackers, yet none of them can be safely relied upon to start and play well next season. Meanwhile, at running back they only have two players under contract: the one-dimensional Legarrette Blount, and undrafted second-year player Mossis Madu, who played about a dozen snaps last year.
With the players still on the board, the Bucs have some options here. Nebraska linebacker Lavonte David is a popular pick for the Bucs, but I'm taking a different route: running back Doug Martin out of Boise State. The Bucs need a running back who can help Legarrette Blount, who can provide them with security if Blount goes down and, most importantly, who can function as a third-down back.
Doug Martin can do all of that. He's a versatile running back who can run downhill with power, has the speed to get to the edge and has enough wiggle to make a man miss in space. He's also an experienced pass-catcher, although his pass-blocking needs to improve as with most drafted players. I've seen people compare him repeatedly to Ray Rice, Greg Schiano's old college running back, and that comparison certainly bodes well for him. In short: Martin's a complete running back who could carry the load if needed, but would at the very least be a very good complement to Legarrette Blount.