I get why Dallas would move on. They've got Heiskanen getting paid to anchor their D corps, they can't really afford to pay Klingberg what he could get on the open market. He's also at that awkward age where you're pretty much guaranteed to be signing a nightmare deal...it's just a matter of how many years of it are a catastrophe vs the productive years remaining.
But i also get it from Klingberg's perspective. You know at this point that you're not part of the Stars future plans, and they can't afford to pay you what you can get elsewhere. They're not really going to win a Cup this year, so it makes sense to ask for a trade to somewhere that's either a contender, or can afford to get that big contract signed and have the ink dry before the season ends (and you risk any sort of derailing injury). Or both.
It also works out for the Stars in the end probably, as they should be able to get something pretty decent, even for a rental Klingberg. He's not a guy you're going to want as your #1, but he's one of the more gamebreaking offensive defencemen in the league still. That's a huge differencemaker for the playoffs, if you can play him appropriately. He'll contribute a ton of minutes, and a ton of points. And his cap hit, especially as a deadline acquisition is basically nothing, for what he offers. So he's easy for pretty much anyone to squeeze into the lineup even a bit before the deadline.
Just sorta reads like a semi-mutual parting of ways to me.