Why do people tend to use negation of negative statement

To expand on what Weather Vane said in their comment, there's a variety of shading of meaning that could be going on.

"He is not unhealthy" doesn't mean "he is healthy." In this case, there's a bit of a fine line with the meaning of "healthy", but that's for medical people to define, not English. A person can be healthy and still feel physically bad.

In Weather Vane's example of wealth, there's a much broader range to work with. Let me explain.

"I'm not poor" can be used by different people and mean different things in different contexts. A businessperson can joking say "I'm not poor" when asked about their earnings to mean that they are reasonably well off, possibly actually rich, and are declining to answer the question specifically.

A poor person saying "I'm not poor" might be said defensively, if you point out their meager paying job, low cost vehicle that's in disrepair, shabby clothing, poor grooming habits, etc. In this context, they could be defending their honor and ignoring the potential fact that they actually are poor.

An "average" person with a reasonably decent job, a family, car(s), debt, home/apartment, etc. might say "I'm not poor" to say that they are living comfortably, but not really gaining ground financially. They might also end that same sentence with "but I'm not rich either" to accentuate that they aren't really making grounds or to put off the idea they have expensive tastes.

A couple phrases I hate are "you're not wrong" and "I don't disagree". What the person is often really saying is that they can't admit that someone else is correct or that they agree with a potential competitor. Both of these can be used as a way to end a conversation.

It's perfectly OK to say "you're not wrong, but you're not totally correct, either". This is not only a complete sentence, but also a complete thought. This allows someone to be partially correct. Also, saying "I somewhat/partially agree" is less ambiguous and more conducive to continuing a conversation.

Saying that someone is "not not guilty" involves legal definitions. In law, there's not really such thing as "innocent". Laws are set up so that someone is guilty or not guilty, so if someone is "not not guilty", it means they are guilty.

As you can see, the context of the statements are entirely necessary to define what a negative, double negative, and other negations can mean. It goes to show that English is about as clear as mud and why even native speakers have problems with it's use.

I hope I helped, but I wouldn't be surprised if I didn't.

It's an example of litotes, an understatement, although the precise interpretation may depend on content. See the definition of litotes.

a figure of speech and form of verbal irony in which understatement is used to emphasize a point by stating a negative to further affirm a positive, often incorporating double negatives for effect

[Wikipedia]

Because talking about the negation of something is not the same as talking positively about the opposite. Even if the "something" itself was expressed as a negation of something to begin with.

In the first instance you are emphasizing the negation. In your example, the phrase "He is not unhealthy, ..." may continue to talk about avoiding unhealthy habits or supporting a healthy lifestyle, all for the sake of not falling ill. Whether he is healthy then, or "just fine" is left open and deemed not important.

The second instance would focus on achieving the positive statement. The phrase may continue similarly than above, but assume that he really is (or feels) healthy as a consequence. In that sense it's a stronger statement about the outcome, removing the grey zone of whatever may be between being healthy and having avoided falling sick.