Even people who believe in objective truth usually have some understanding that what they see as objective truth is not universally held to be true. A response to a competing “truth” is usually an escalation into a war of words, if not more, since there can be no compromise on “truth.”
Thus people use this idea of a “personal” truth to keep things at a civil level of discourse. Even though we don’t actually acknowledge there is any truth to someone else’s truth, we do acknowledge (when we speak this way) that someone else has another way of seeing things that they think of as “truth.”
In this way, we try to keep things civil and allow competing and contradictory truths to stand side by side. It is a way of expressing respect for the humanity of someone else, even if you think they are totally wrong.
So in acknowledging another person’s truth, you are not actually acknowledging that truth is true. It is a social fiction that allows us to keep from beating the crap out of each other. We can continue to say that we hold this truth without having to say that the other person’s truth is stupid, ignorant and unbelievably dumb. “Well, that’s your truth.”
Of course, if you are a relativist, then you reject the notion of truth as an objective thing. It is something that is the best interpretation you have of the best evidence you have on hand. You understand that there are other interpretations of the evidence, and that other people have access to different sets of evidence. Therefore it is natural that there may be discrepancies in the way people see the world.
Whether objectivist or subjectivist, this idea of different truths is often used, in my opinion, as a way of smoothing troubled waters and allowing discussion to continue at a slightly less heated level. It is a rhetorical technique, more than some kind of principled philosophical outlook.