What is a field ? What is magnetic field or other fields made of or what it is, How do u define it (To my knowledge field is thought to exist without a proof and theories are built over it) ?
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I am starting with your argument:
If you've problem with this mathematical modelling of physical problem, ask yourself, "Do you know what Energy is?" Man, its all that number to describe how things happen in the world around us.
The field is actually distribution of physical quantity on each point of spacetime (you can say only Space in classical sense, but it'd pose major problems while describing today's complex fields).
But, why field when we can measure physical quantities on any point without it?
A field is simple a function of space(-time) that assigns some value, vector or pretty much anything, to a certain point in space(-time).
Nothing fancy really. A normal function like $f(x) = x^2$ can be viewed as a field that assigns a real value to the space $\mathbb R$.
We call the field by the type of element it assigns. So a Vector field assigns a Vector to each point in space.
You can define your fields over whatever elements you wish. Heck, you could create an apple field, by assigning a certain number of apples to each point in space.
It's pretty much the same as the mathematical answer. The only difference lies in how you interpret these fields.
So a Vector field, that assigns a vector to each point in space can be viewed as a magnetic field.
For example, think of a ball (the earth). Now, think of a vector field on this ball, i.e. some function that assigns a vector to each point. We say this vector field is smooth, if the vectors of two nearby points only differ by some small $\epsilon$-vector. Think of the vectors as hairs on the ball. Smoothness then just means that the hair looks tidy and two neighbouring hairs have almost the same direction.
You can reconstruct the earth magnetic field by choosing an appropriate function. Interestingly, if you do this, we have a mathematical theorem that says, that such a vector field must essentially have (at least) 2 poles (or a pole of multiplicity 2). In our case, the North and South pole. Or in short "You can't comb the hair on a coconut". This theorem is called the Hairy ball theorem.