One reason to believe Mordell's conjecture is Chaubaty's proof of it (from 1940ish) in the case where the Mordell-Weil rank of the Jacobian is less than the genus of the curve. This basically builds on the idea from Milne's notes mentioned in George Lowther's answer, but instead uses the $p$-adic Lie group structure of the points of the Jacobian in a non-archimedean completion of the number field. You can read an explanation of the method in section 6 of these notes by Brian Conrad.
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