Weil's proofs for curves and abelian varieties essentially use special cases of the standard conjectures, and the framework of the standard conjectures (like many other conjectures of a motivic nature) is suggested by trying to generalize the abelian variety case (or if you like, the case of H^1) to general varieties (or if you like, to cohomology in higher degrees).

Deligne's proof for K3 surfaces uses a motivic relation between K3s and abelian varieties (which is most easily seen on the level of Hodge structures) to import the result for abelian varieties into the context of K3 surfaces. This is not so different in spirit to Manin's proof for unirational 3-folds, except that the relationship between the K3 and associated abelian variety (the so-called Kuga-Satake variety) is not quite as transparent.

Weil's proofs for curves and abelian varieties essentially use special cases of the standard conjectures, and the framework of the standard conjectures (like many other conjectures of a motivic nature) is suggested by trying to generalize the abelian variety case (or if you like, the case of H^1) to general varieties (or if you like, to cohomology in higher degrees).

Deligne's proof for K3 surfaces uses a motivic relation between K3s and abelian varieties (which is most easily seen on the level of Hodge structures) to import the result for abelian varieties into the context of K3 surfaces. This is not so different in spirit to Manin's proof for unirational 3-folds, except that the relationship between the K3 and associated abelian variety (the so-called Kuga-Satake variety) is not quite as transparent.

[Added, in light of the comments by Donu Arapura and Tony Scholl below:] In the K3 example, it would be better to write "a conjectural motivic relation ... (which can be observed rigorously on the level of Hodge structures) ...".