First off, I think you got the actors the wrong way round.
A また太っちゃった。
I ended up getting fat again.
B あまいものばかり食べているからだよ。
That's because you eat nothing but sweets.
Can you remove からだ? Grammatically you can, but it wouldn't sound natural in the same way that this English exchange would sound slightly awkward:
A) I ended up getting fat again.
B) You eat nothing but sweets.
B's reply doesn't feel like part of the conversation does it? It just sounds like a random statement of fact.
Edit: I've just realised your problem. You thought からだ was 体 meaning body. But it isn't. It is から meaning 'because' plus the copula だ.