A runner in Mount Prospect may be told that a metatarsal stress fracture is just "part of the sport." The documented explanation is more specific than that.
Repeated stress, applied the same way
The direct answer is repetition. A stress fracture comes from stressing the foot in the same way again and again, and running is one of the activities that does exactly that.
What the source says
MedlinePlus states: "Stress fractures are caused by overly stressing the foot when using it in the same way repeatedly." The page is its aftercare guide for metatarsal stress fractures.
MedlinePlus also lists people who do "activities that put a lot of pressure on their feet, such as running, dancing, jumping, or marching (as in the military)."
Running sits in a list with dancing, jumping and marching. What they share, in that wording, is a lot of pressure on the feet.
Reading further
The Cleveland Clinic has a page on stress fractures, their symptoms, causes and treatment. AAOS OrthoInfo covers stress fractures of the foot and ankle. One PubMed Central case report asks whether bone cement for second metatarsal stress fractures is safe, which is a reminder that titles can pose a question rather than answer one.
What to ask next
If you run and have foot pain, ask the clinician which activity pattern they think is loading the foot. Ask how that pattern is meant to change while the foot is treated.
How the claim reads against the evidence
"Part of the sport" is a label, and the evidence replaces it with a mechanism. Repeated pressure on the foot is the documented cause, and it points to something a runner can talk through with a clinician.