One of the definitions of なる (成る) is to be completed (完成する), or to succeed (成功する), and thus carries the implication of "good". Consequently, ならない then means not completed; not successful; "it won't become (good)". As in English, if you say "That won't do", it begs the question: won't do what? But it has been ingrained in your mind to know that "that won't do" = "that's no good".
So to break it down into parts, 〜なければ means "if you don't 〜", and as we just said, ならない by itself means "no good" or "that won't do", you get "That won't do/be good if you don't 〜". Thus implying that you must do it.
This is why there are multiple ways to express having to do something. Instead of ならない, you can substitute いけない or だめ, both of which also express "no good"/"bad". And instead of 〜なければ, you can substitute 〜なくては. The translation changes slightly from "If you don't 〜", to "Not doing 〜", but gives the same overall semantics. Compare
- 全部食べなければならない → It's no good if you don't eat all of it
- 全部食べなくてはならない → Not eating all of it is/would be no good
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