Let's take an example of magnetic field lines due to a current flowing through a wire. For the sake of the example, assume the wire of infinite length. Now, the magnetic field at any point radial to the wire can be derived. My question is, why are they lines and not a disc area? Why are the iron filings arranged in concentric lines and not a disc with varying thickness, from more thick to less as the distance increases radially?
First, I believe our question is really about how the iron filings are arranged, not about why magnetic field lines are lines. Field lines are abstract concepts which are by definition lines rather than disks (2d objects, or 3d if you allow thickness).
Why does a bunch of iron fillings form a pattern that traces the abstract magnetic field lines, rather than just clustering into a 3d object? This is a nontrivial question.
The answer comes in three conceptual (not chronological) steps. First, the magnetic field magnetizes the iron fillings into a bunch of tiny magnets. Let's just treat each of them as a small round particle. Second, these tiny magnets exert forces on each other. One can show that with some perturbation, the forces prefer them to form a chain with head-to-tail (north-pole-to-south-pole) alignment, rather than a shoulder-to-shoulder configuration. So the powder-like iron filings will form a bunch of tiny needles that are elongated in one direction. Third, each needle behaves like a tiny compass, that points to the direction of the external magnetic field, due to a force known as the magnetic dipolar force. This pattern of needles traces the magnetic field lines.
My question is, why are they lines and not a disc area? Why are the iron fillings arranged in concentric lines and not a disc with varying thickness, from more thick to less as the distance increases radially?
The arrangement of the iron filings depends on how many filings you place and where they end up. The purpose of the filings is to see the direction of the field lines, not their density, because the density we would see would just depend on how many filings I put down. For example, if I put one filing close to the wire and then dump a bunch of filings farther away, by your measure you would think the field is stronger farther away from the wire, and this is not the case. For an accurate representation of the field using filings you should have a uniform filing density everywhere. Then the filings will show you the direction of the field at each point in space.
In terms of why we draw field lines as lines is because by definition field lines are lines. They just indicate the direction of the field at some point in space. Of course, there is a magnetic field at all points in space, so we choose certain points to draw rather than draw everything (which would, in fact, be seen as a disk). Usually diagrams like these are drawn such that a larger density of lines indicates a stronger field, so in this case you would get what you ask: more lines to fewer lines as we move out.
Of course, you can represent the field in other ways. For example, you could shade in a disk whose shading indicates the magnitude of the field at that radial distance. It all just depends on how you want to visually represent everything.