What's the difference between these two questions:
Why they chose football?
Why did they choose football?
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The first one is not grammatically correct, for a question; the second one is the correct form of that sentence.
You are right in pointing out that "W" questions require that auxiliary verb, 'did'. (or more generally, 'to do') to connect a "W" (or how) question to another subject.
In the examples below, the subject that 'to do' is connecting the "W" question to is surrounded by asterisks.
Who did *you* meet? Why does *that* happen to this computer? Where do *you* go when you leave the house? How does *a magnet* work? When did *the kids* come in?
The first sample, "why they chose football", is a noun phrase referring to their reason(s) for choosing football.
The second sample, "Why did they choose football?", is a question asking for their reason(s) for choosing football.