Now, to cavalierly assume a causal connection between this tension and LA's failure to make the playoffs--as Brooks goes on to do--is stupid. The Kings posted excellent underlying numbers all year long and missed the playoffs primarily because of bad luck, especially in the shootout and overtime--see this piece for more on that. So unless Kings shooters were deliberately botching their shootout attempts to protest their coach, I think the "Kings-missed-the-playoffs-because-they-hate-Sutter" theory doesn't hold up. Furthermore, the Kings' Corsis actually improved significantly as the season went on, which runs counter to the notion that the Kings stopped trying hard for their coach down the stretch (or the theory that physical exhaustion got to them, for that matter).
You can make of this report whatever you want. A lot of details are missing and we have no idea which players felt this way, or for how long, nor do we know how this situation ultimately resolved (maybe the players and Sutter discussed it the next day and amicably put it behind them, maybe not, we have no idea). But I will say that I don't care much, because I think stuff like this is really common in sports. It's a coaches job to drive his players and to be abrasive and unpleasant in an effort to get more out of them than they'd otherwise give. There is naturally going to be tension. It's really common for reports of that tension to surface once a prominent team loses, because it makes for a great narrative. But it probably goes on in most locker rooms.
I'll share a quick story: after the 2012 Baltimore Ravens suffered a particularly lopsided loss, coach John Harbaugh tried to have the team undergo an especially grueling practice in pads the next week. The players simply refused, a situation one of them described as a "near-mutiny." That same 2012 Ravens team, if you didn't know, won the Super Bowl a couple months later.
The point is, some amount of friction is just the reality of coach-player interactions, and it usually does not matter much to wins or losses in the end. We don't have all the details so I can't be totally sure about that, but this is probably much ado about nothing.
Update: Dennis Bernstein reports that LA players completely reject Brooks' account. So maybe this is even more of a non-story than I've made out.