kowrip said:
Like others have said, everybody plays to win. Other players should understand this and have most likely been in your spot before. Still, I believe there is no need to gloat or be nasty. I usually say something like "Good game" or "It's been nice playing with you". If a good player gets eliminated with, say an over pair against as set, I might add something like "There's no shame in going out that way".
I really like this general response, and would like to add that if the person seems especially upset to be eliminated (more likely in a deep-stacked, post flop all-in where a lot of people would probably describe the situation as you having “outplayed” your opponent, as well as in bigger tournaments), then just taking a step back from the table to give them time to gather their stuff and leave without forcing them to interact with you can be the kindest thing I think.
In less intense situations, when I can genuinely say it, I often tell the other person that they played great, and it was a cooler. Or, I would’ve called there too, for example.
As a last note, not gloating or displaying poor sportsmanship extends to social media! I always think less of pros that I see mocking people they knocked out, even when it’s not blatant. If you catch yourself wanting to berate someone once you’ve already taken their chips, in reality it probably means that the berator has waaay more work they need to do on their own game!