I think the reason so many people are having difficulties with this sentence is that the use of "so" in the English translation is not standard. All od the other options - "hence", "and so", "therefore" - make more sense. "So" by itself suggest the purpose or intent of eating the vegetables is to become strong (ie, Polish "aby", not "więc"), and would require future tense in English.
While the meaning is roughly the same ("eating vegetables makes me strong") the two sentences convey different things:
"I eat vegetables so I am strong" (="Jem warzywa, więc jestem silna") expresses an implication.
"I eat vegetables - that's why I am strong" (="Jem warzywa - to dlatego jestem silna") focuses on the cause, the part "I eat vegetables" is emphasized.
IMO the first one is more universal; the second would be used e.g. as an answer to the question "Why are you strong?" (="Dlaczego jesteś silna?").
PS: remember that if you are male (and are speaking about yourself) you have to use "silny" instead of "silna".