What is a nature of a magnetic field

Surface level answer: because electromagnetism.

It happens that for 2 currents flowing in one direction it’s energy-wise more stable to be closer to each other - and wise versa.

Deep level answer: nobody knows why exactly is that.

Because electromagnetism is just a model to describe what we see with some math approximation.

Maybe it’s just we were lucky to be born in the universe with such properties and able to talk about it.

https://en.wikipedia.org/wiki/Fundamental_interaction

There are no magnets only electricity.

When electricity moves fast (close to the speed of light) relativistic effects occur which cause electric force to appear like (what we call) magnetic force.

So the real question is "Why do electrons use to react to each other?"

And the answer is, no one knows, no one can know, the best we can do is describe the behavior.

Afaik, science doesn't know this yet. I've seen an interesting theory that noticed that if we take the navier-stokes equation of an ideal flow, insert the lorenz factor there, we can relatively easily derive both magnetic (maxwell) and gravity equations. If that guess is correct, the nature of magnetism is a flow of 'water' that assumes curious shapes, forms stable structures (particles) and binds them together. The nature of that 'water', however, will remain a mystery. Problem is such theories are impossible to verify today.

The basic answer is that nobody knows. I've been asking this question since I was a very young kid when I saw a magnet for the first time and many, many decades later I still have not found a satisfactory answer - nor anyone who can convincingly answer the question!

Physicists are forever talking about the beauty of the Standard Model of particle physics and how it can be explained in terms of QFT - Quantum Field Theory and so on but at the most basic level they still don't know what's actually in that field 'stuff' that 'glues' your fridge magnets to your fridge.

Have you noticed how this subject is so often avoided by physicists? Similarly, they also avoid the question 'what exactly is 'inside' a force that makes it do things?'.

Edit: several years ago Nobel prize winner Frank Wilczek was asked by New Scientist magazine what a field was and the best answer he could come up with was that it is a kind of matrix. At the time I thought this was a totally unsatisfactory answer.

Magnetic fields are an effect of looking at moving electric charges from relativistic points of view. There is always at least one observer that sees only an electrical interaction because of their reference frame.

In that reference frame, there are no magnetic fields that influence motion, just electrical charges attracting and repelling.

We don't know. Seriously - other commenters have elaborated more on the topic. There's a related rap piece slash Internet meme that is the first thing that came to my mind: https://www.youtube.com/watch?v=8bhYMnHb5JY

One of the world's best expert and teacher said:

"It's just one of those things that you have to take as an element of the world.... I can't explain the attraction in terms of anything else that's familiar to you."

https://m.youtube.com/watch?v=MO0r930Sn_8