What TSR stands for (from my understanding) is "terminate and stay resident". It allows programs or other things to start and stay in memory in a "ready to go" mode... basically sitting their till its told to do something.
Im by no means an expert, but Im thinking it might slow down the games because it takes up a little bit of memory. Some games would use more memory then others, and I'm assuming the games that are slowing down with this enabled use more memory than the ones that dont slow down.
Back in the "DOS days, I used to have a game cheat program on my computer years ago that you could custom make cheats with. So if youre playing prince of Persia, and you have 4 "marks" of life, you would hit the ~ key, and the program would open. You would tell it to search for the value of 4, and then go back to the game. When that value goes up or down, to either 3 or 5, you would go back into the program by hitting ~ and type in what the value is now. It would search for what lines in memory changed. You could then edit this line so the game would reflect 20 marks of life, or you could freeze it so that you would never lose life, or gain life, etc...
Thats an example of a "terminate and stay resident" program.