I'd guess it's something between "I'm pretty sure I'm right but I don't want to argue about it with you right now" and "fuck off brah".

Depends on the context, obviously, but my first interpretation would be "My values are not your values". In popular usage "truth" means more than empirically proven facts about the objective reality -- e.g. people routinely call "truth" what they believe not only in the descriptive but also in the normative sense.

I would recommend making clear two separations: between descriptive ("US economic growth has been slow recently") and normative ("We need to accelerate the US economic growth"); and between facts ("The US GDP grew by 2.4% in 2015"), preferences("Fighting inequality is more important than gross economic growth"), and forecasts, often conditional ("We can accelerate the economic growth by cutting taxes").

While it is safe to assume most people will understand that statement in their own way making me even doubt if I should write a response, I think some words can be said by me on the subject.

The statement is an affirmation by the issuer to communicate that it is aware that what it has understood of the world can be different to what you have understood of the world. The level of which the statement was meant depends on what it was thinking at the time as it could have logical grounds or grounds rooted in the realm of physics.

The desired effect can either be to initiate conversation regarding that idea or as a simple recycled move to limit the way the conversation is going or the way it can go.

They might simply mean "your apparent interest in truth is not an interest in matching reality, but in matching your feelings or something like that." They may or may not be right depending on who you are.

I expect that "truth" in this context just mean "opinion what the truth is" which can not be easily verified nor refuted and despite of it both debaters are pretty confident in their opinions. In this case a response can be something like "I would prefer word "opinion" but I hope we understand each other ;-)"

For more accurate response I need some examples. I don't realize coming across this phrase in a discussion.

My job is selling books. On the one hand, I should sell as many and/or as expensive as possible books; on the other, since customers keep coming back, I should (if they explicitly ask my opinion, or if I see them hesitate) offer them deals that would be okay to them (I don't aim higher - there are time constraints, after all). Taken together, this means I have to make assumptions about what they want and how much they want it (it's actually interesting - listening to imprecise words, reading faces, trying to remember if we have something without looking it up, etc.). Recently, a lady asked what I would consider a good present for a 11 y.o. child. 'How about Tom Sawyer?' 'Oh no, I mean, for a girl.' Now, I had originally offered 'Tom Sawyer' exactly because I did not know the child's gender, but her answer made it look like I had assumed that she was talking about a boy. I could have corrected her, but instead I pointed out 'Ann from Green Gables' and we parted (more-or-less) satisfied. (Obviously I'm still smarting.)

The point is, we could have argued about both our assumptions and our conclusions, but it was not worth it, and it did not matter much. Who of us was right?

Looking for a single answer that applies to all cases seems like a clear mistake. That said, your interlocutor probably does not think in terms of expected value, so they may have an epistemology that attempts to do the same job.

In my experience, people appear to have informal, relatively fluid and vague concepts for things that also have formal, precise and rigorous definitions/expressions in systematic thought, shall we say. Truth appears to be one such thing, as I am sure others have noted here. When someone speaks of "my truth" there could be a few things implied or confounded within that declaration

1. What "feels" true to me right here, right now
2. I can see the arguments, but the chain leads to a core belief of mine that simply cannot be wrong given my worldview, identity etc, therefore it may seem true to you, given what you know, but it is not true to me given what I know to be also true
3. As others have suggested, this may be a conversation-stopper because the argument is emotionally upsetting, or the other side suspects your motives or reasoning but does not want to engage, or they simply do not care for the emotional and mental energy involved in adjudication of such matters as truth (as defined as beliefs matching some presumed objective reality).
4. People's reasoning often appears motivated, utility-driven (be it status, or actual rewards or whatever), ergo they may be redefining truth in terms of what would maximise their expected utility in the context in question...so, for example if you, a sceptic/atheist are debating a religious person in the company of their fellow believers, they have more utility in winning or appearing to reach a noble impasse with you than re-negotiate their beliefs in public. Given how a lot of people do not like to admit to being wrong (and this includes everyone, at some point, and requires good metacognitive discipline to overcome), even if it is a two-person conversation, if the topic is one that requires "truth" to be brought into the picture (well, "Pikachu is ridiculous"...could be the topic, for all I know...in which case, there are very few truthful propositions to examine...), the motivation to not lose face, or not yield may be strong.

The rational response to such a comment is to issue oneself a firm "nolle prosequi" and exit the conversation politely*. I have no idea what the conversation was about, so I cannot know the truth of things ;)

*if the topic happened to be one with sufficient fogginess in the real world , re-examining one's own beliefs would be a necessary step - heck, I'd re-evaluate anyway, just as a sanity check.

Quite possibly. But the first is more dangerous and so there is more reason to pay attention to it.

Can someone help me dissolve this, and give insight into how to proceed with someone who says this?

You don't, they just don't want to talk about it. Some people can sadly not be saved.