A magnet's field is not a picture. It is a rule, at every point in space — and iron filings are the only honest way to see it read out.
The bar is two point poles of opposite sign. Each pulls with an inverse-square hand, and every splinter of iron reports the sum of the two — which is why the classic loops curve out of one end and dive into the other.
Each filing is a tiny compass. It cannot show which way is north — only the axis of the field where it sits. Ten thousand of them, each turned to its local vector, and the invisible geometry becomes a photograph.
Summon another magnet and the two fields add. Opposite poles knit their lines together; like poles shove apart and leave a dead null between them — a point where the field cancels to nothing.