All effects with a fixed duration measurable in rounds end just before the beginning of one of your turns. (Otherwise, their duration would be too short to be a full round, or a full 2 rounds, etc.)
We can see this by looking at a duration of 1 round, and applying the same logic to the case of 10 rounds.
A duration of 1 round, measured in turns, is “each participant in a battle takes a turn” (PHB, p. 189) — or put another way, one turn each for you and everyone else. Since you enjoy the benefits of an effect like Rage on the turn you activate it, that's one turn of effect for you; everyone afterward also “enjoys” the effect for their turn after you, until the end of the turn of the creature just before your next turn. When your next turn begins, you would be enjoying the effect for a second turn, therefore the 2nd round of the effect begins as your 2nd turn begins.
Thus the dividing line between the rounds, for an effect you enjoy on its first turn of activation, is just as/before one of your turns begins.
(There are a few exceptions, but these explicitly say when they end in relation to your turn. Most are single-round spells that alter your next action, which explicitly extend to the end of you next round so that they're not useless — you can think of effects like that as beginning to be “enjoyed” by you and everyone else only after the end of your 1st turn.)
Now we can extend this out to 10 turns, to see that Rage ends just before your 11th turn begins. For simplicity, we'll call the turn an effect starts “your 1st turn”, no matter how long combat has been.
A duration of 1 round begins on your 1st turn and ends just before the beginning of your 2nd (next) turn. Otherwise, it would not be a full round long.
A duration of 2 rounds begins on your 1st turn and ends just before the beginning of your 3rd turn. Otherwise, it would not be a full 2 rounds long.
A duration of 10 rounds begins on your 1st turn and ends just before the beginning of your 11th turn. Otherwise, it would not be a full 10 rounds long.
So in the case of rage, it ends just before you take your 11th turn, preventing you from enjoying 11 rounds of rage but allowing you to enjoy the full 10th round of your rage, including during everyone else's turns.