This question is testing PRONOUNS,

If a sentence begins with an opening modifier or dependent clause containing a pronoun (e.g. "Although they are..."), that pronoun must be referring to the SUBJECT of the main clause that comes after. So in the case of "Although they are..." the subject must be PLURAL to refer to "they."

(A) they are crucially important, a person's total calorie intake is only one of the many factors that determine if their weight will increase or decrease
- "they" is presumably trying to refer to "calories", but the subject is actually "intake" - incorrect pronoun.
- "their" is incorrectly trying to refer to "a person."

(B) it is crucially important, a person's total calorie intake is only one of the many factors that determine whether his or her weight will increase or decrease
- "it" correctly refers to "intake"
- "his or her" correctly refers to "a person"

(C) it is a crucially important factor, a person's total calorie intake is the only one of many that determines whether his or her weight will increase or decrease
- "it" correctly refers to "intake"
- "his or her" correctly refers to "a person"
- however, switching the position of "only" changes the meaning. We're now saying that it is the ONLY factor that matters. That wouldn't make sense with the contrast "although important... it's the only one"
- also, "one of many" is unclear. One of many what? It's much clearer to say "one of many factors" as in B.

(D) crucially important, a person's total calorie intake is only one of the many factors that determines the increase or decrease in their weight
- "their" is incorrectly trying to refer to "a person."
- "(one of the many) factors" - here, "factors" is the subject, not "one." The verb should be plural: "determine."

(E) it is crucially important, a person's total calorie intake is the only one of many factors that determines the increase or decrease in his or her weight
- "it" correctly refers to "intake", and "his or her" correctly refers to "a person"
- same "only" issue as in C

The answer is B.