Just because the mother happens to be beautiful doesn't necessarily mean her daughter will grow up to be a beauty.
Literal
Mother [subject-が] happens-to beautiful [because-からと言って], necessarily that daughter [subject-が] grow-up-[sequential-て] beauty become [quotative-と]-say thing [at/topic-には] not-become
~からと言って...とは限らない is split here into ~からと言って...という事にはならない, a close variant meaning 'it doesn't follow that.' たまたま = 'happens to / by chance.' 美人になる = 'to become a beauty.' The sentence is a lesson about logic: correlation ≠ causation, Japanese-style.