'I don't disagree that...'

“I don’t disagree” doesn’t exactly mean “I agree” It’s more like “although I don’t completely, enthusiastically agree, I see your point”

“I don’t disagree that…” would usually be followed by something that contradicts or corrects the rest of the claim. E.g. “I don’t disagree that homelessness is an issue, but I think there are better courses of action than what you proposed.”

"You might assume that I disagree with it, but, in fact, I don't." Or "I'm reluctant and unenthusiastic about agreeing with this, but I must admit, I don't find it wrong."

You would use this structure when trying to highlight contrasting opinions on something. For example A: I think that we need to have harsher mandatory minimum sentences for certain crimes, so criminals will think twice before committing any crimes. B: I don't disagree that people should be punished for crimes, but it has been proven that mandatory minimum sentences are not effective deterrents of crime.

Can be similar to "You're not wrong." Which is a bit like "You said it, I didn't." Or "I might have said that differently, but yeah pretty much."

"I don't disagree" doesn't necessarily mean you agree. It could mean you don't have an opinion either way. 

“I don’t disagree” is a way of expressing that you may partially agree, or unenthusiastically agree. Usually you’ll probably find that an expression like this does not stand alone in a discussion. You’ll likely get some sort of elaboration on the opinion. “I don’t disagree, but…”

Usually, "I don't disagree" is followed by " . . . but . . . " The speaker is going to actually disagree with certain parts of your statement or argument, or maybe just add some nuanced aspects of it. "When you say the sky is blue, I don't disagree with you, but many people further describe the sky's color with words that indicate slight changes in shading and hue, sometimes for poetic effect."

It’s usually used when talking about a complex situation with numerous points. Person 1: “I think that Taylor Swift makes great music, and that she is the best artist ever.” Person 2: “While I don’t disagree that she makes great music, she definitely isn’t the best artist ever.”

"I don't disagree" doesn't mean you agree. You could be ambivalent, for example. Usually, it indicates someone hasn't fully decided, and may ask for more information, clarification. "I don't disagree that they landed on the Moon, but why haven't they been back in 50 years?"

I don't disagree and I agree are similar. The value/use of "don't disagree" is in a context where it *might* sound like whatever you say next is contrary (or perhaps that part of it is contrary so you need to differentiate). It's really cold outside, so you should stay inside. I don't disagree that it's cold outside, but I can go outside with a coat. If you said, "I agree..." then the listener might think you agree with the whole thing, but you only agree with part.

I usually see it more along the line of 'I don't disagree' = 'you're not wrong (about X)'. In a sentence, 'I don't disagree about X, but I don't think you understand how important Y is.' If you were talking to somebody learning about nutrition, 'I don't disagree beef is a great source of protein, but fruits and vegetables provide important vitamins and minerals also.'

Similarly to “you’re not wrong”, it signifies only partial negation.

So it's to say you're not wrong, but you're not right either. Usually 'I don't disagree' is followed by but[counter point/argument] it's a scale 'I don't disagree' falls just above 'maybe your right' and before below 'I agree.'

It's called litotes

“I don’t disagree” means just that. Just like “not guilty” doesn’t necessarily mean innocent.

Because you don’t agree strongly or you only accept a point of the argument, not the entire argument.