In fact the case of curves over finite fields had been proved by Weil himself, finishing the project started by Hasse's theorem on elliptic curves over finite fields. The conjectures were natural enough in one direction, simply proposing that known good properties would extend. Their interest was obvious enough from within number theory, since they implied the existence of machinery that would provide upper bounds for exponential sums of interest in analytic number theory.
What was really eye-catching from the point of view of other mathematicians was the proposed connection with algebraic topology. Given that finite fields are discrete in nature and topology speaks only about the continuous, the detailed formulation of Weil (based on working out some examples) was striking in the way that it suggested that geometry over finite fields should fit into well-known patterns relating to Betti numbers, the Lefschetz fixed-point theorem and so on.
Weil himself, it is said, never really tried to prove the conjectures. The analogy with topology suggested that a new homological theory be set up applying within algebraic geometry. This took two decades (it was a central aim of the work and school of Alexander Grothendieck) building up on initial suggestions from Serre and others. The rationality part of the conjectures was proved first, by Dwork, using p-adic methods. The rest awaited the construction of étale cohomology, a theory whose very definition lies quite deep. The proofs were completed by Pierre Deligne, roughly speaking using a painstaking induction on dimension.
The conjectures of Weil have therefore taken their place within the general theory (of L-functions, in the broad sense). Since étale cohomology has had many other applications, this development exemplifies the relationship between conjectures (based on examples, guesswork and intuition), theory-building, problem-solving, and spin-offs, even in the most abstract parts of pure mathematics.