Break-in is treated as a period of endurance before a shoe becomes comfortable. Something specific is happening to the materials, and it only occurs in certain constructions.
Leather deforms permanently under load
Leather is a fiber network that yields when stressed, and beyond a certain point the fibers rearrange rather than springing back.
Body heat and moisture from the foot soften the fibers further, which allows the upper to take the shape of the foot pressing against it.
That accommodation is permanent, which is why a broken-in leather shoe stays comfortable and why it never returns to its original dimensions.
The insole compresses to a footprint
Traditional welted shoes contain a leather insole with a filler layer beneath, and both compress under the pressure points of the specific foot wearing them.
Over time the compression forms a shallow impression matching the wearer's sole, which distributes load more evenly than a flat surface does.
This is the main reason well-made shoes become more comfortable with age, and also why a secondhand pair may never fit the second owner well.
Synthetics do not break in
Most synthetic uppers and foam midsoles are elastic within their working range, meaning they return to shape rather than taking a set.
A synthetic shoe that hurts on the first wear will usually still hurt on the twentieth, because nothing in the structure is capable of adapting.
Knit uppers are the exception in a different sense, since they conform immediately rather than gradually and require no break-in at all.
Stiff soles flex at a fixed point
A leather outsole is rigid when new and develops a flex crease at the point where the foot bends, which then becomes the permanent hinge.
If the shoe is slightly too long, that crease forms ahead of the natural flex point of the foot, and the mismatch never resolves with further wear.
This is why length errors are less forgiving than width errors, and why a shoe that is too long cannot be broken in successfully.
Where break-in is the wrong explanation
Pain across the top of the foot, numbness in the toes or heel slipping severe enough to blister are fit problems rather than stages of a process.
A cobbler can stretch leather locally over a specific pressure point, which addresses a genuine hot spot faster and more reliably than wearing through it.
Persistent foot pain that continues after a change of footwear warrants a podiatrist, since it may reflect structural issues that no shoe adjustment will resolve.