You do need to be reasonably comfortable with basic algebra, if only so that you don’t worry about it; you do not need trigonometry or calculus. (If I had my way, high schools would replace most of their calculus courses with discrete math and elementary statistics.) What you chiefly need is the ability to follow logical arguments. Oh, and you need to be aware that you’ll often be dealing with exercises that are much less routine in nature than those in the math courses that you’ve had so far. Most of the mathematical tools that you’ll use in an introductory discrete math course are pretty simple; the trick is generally figuring out which ones to use on a particular problem and how to use them.
While it is true that Weil appears to require that his reader appreciate Haar measure, I think there is serious risk of over-interpreting this. If anything, an attentive reading of the initial parts of Weil show exactly what one needs of invariant measures and integrals... and it is just the same as needed for Iwasawa-Tate's rewriting of Hecke's treatment of L-functions of number fields.
Rather, I think one might be happiest with a prior reading of a serious (as opposed to easy-intro) book on Alg No Th, such as Serge Lang's. After reading that more-conventional treatment, one will have an idea what Weil is accomplishing by seeming (!) to emphasize measure... as his foundation.
Although Weil's book contains a number of things difficult to find elsewhere, it is also rather quirky, and in a fashion not excessively reader-friendly. Having it be the second more-advanced book one has read on the subject is desirable.
(Systematic reading of whole books about functional analysis or measure-and-integration is surely not necessary, although, yes, of course, such background would provide a "complete-logical" background. It wouldn't explain the number theoretic motivation. And I note that the usual books on "real analysis" seem inordinately devoted to scaring us and making us worry about the legitimacy of doing anything at all... rather than enabling us to do more than we could previously.)
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If you don't know a great deal of abstract algebra so far, maybe "A First Course in Abstract Algebra" by Fraleigh might be a good place to start, as it includes all the prerequisites (groups, rings, fields, linear algebra) as well as a very readable treatment of Galois Theory itself.