I always thought that the field lines drawn in magnetic fields in diagrams were simply visual aids showing the direction of particles at arbitrary points people chose to place a field line. However, when viewing images of iron filings being acted upon by magnetic dipoles, they group together and follow distinct lines. Does this mean that the field lines aren't just visual aids but real physical phenomena?
I think the linked answers don't satisfactorily justify the phenomenology of the iron filings, so here's an attempt to answer that part of the question only.
The apparent lines are a thing ferromagnetic things (like iron filings) do in a continuous magnetic field, not characteristics of the magnetic field.
Iron particles align in a way resembling field lines because the high magnetic permeability of iron makes the iron particles function like tiny bar magnets whose poles are lined up in the same direction as the magnetic field. The magnetic field itself is continuous.
Just like bar magnets, they are most stable when they line up pole to opposite pole.
Furthermore, the strength of the field at the end of the line of iron filings is much stronger than it is elsewhere. This is again because of the iron's high magnetic permeability. The magnetic field of the magnet diverges much less when it is passing$^1$ through iron than when it is passing through air, effectively focusing the field strength to the end of the cluster of filings farthest from the magnet. This is analogous to how a fiberoptic cable focuses light from a source to the end of the cable, although the mechanism is different.
So: the lines emerge as a consequence of the attractive and repulsive forces between opposite and like poles of magnets respectively, combined with the fact that the iron particles are magnetized in the direction of the magnetic field wherever they randomly happen to start out, and the magnetic field being stronger near the outer end of a cluster of iron filings than elsewhere nearby.
$1:$ despite the word choice, no actual movement of the field is going on