Kevin Kiermaier is a center fielder. A very good one, in fact -- baseball-reference gave him 5.0 defensive WAR in 2015, by far the best for any regular CF in 2015 (Kevin Pillar of the Blue Jays was second at 2.8).
So why would the Rays trade him? Same reason, really, they've traded any young player who's become a good regular. The Rays don't have money for a large payroll (their Opening Day 2015 payroll ranked 28th, ahead of only the Astros and Marlins). Eventually Kiermaier will get too expensive for them, though he's not arb-eligible for two more years, so they could hold out for a very good return until 2018.
There's one more issue with Kiermaier -- he doesn't walk much. Even though he had a decent .718 OPS (97 OPS+) in 2015, he only walked 24 times and had an on-base percentage of .298, which is pretty bad for an everyday player. It was also a decline from 2014, when in part-time play (106 games) he had an OBP of .315. Neither of those figures would be good enough to lead off. So if the Cubs did acquire Kiermaier, they'd have to probably bat him ninth, which means Addison Russell likely becomes the team's leadoff man.
Kiermaier does have some power. He hit 25 doubles, 12 triples and 10 home runs in 2015; the triples ranked second in the American League, and his 18 stolen bases (just five caught stealing) gave him a 78.26 percent SB rate, ninth in the A.L.
So what you'd get with Kiermaier is some speed on the basepaths, a bit of power (his .420 SLG in 2015 is decent) and excellent defense in center field. Worth it? And who would you have to give the Rays to get him? Your turn.