This phenomenon is specific to the fields $\mathbb F_q$ and $\bar{\mathbb F}_q$. Roughly, the outline is as follows.

If the Tate conjecture is true, then the homological standard conjecture is true, so the category of motives is semisimple by Jannsen's work. Moreover, the Tate conjecture gives a description of the simple objects in terms of the characteristic polynomial of Frobenius. Then Honda–Tate theory constructs all such characteristic polynomials inside the cohomology of an abelian variety. These are always dominated by Jacobians, so all desired Frobenius eigenvalues occur in the $H^1$ of a curve.

A great general reference for motives is the AMS *Proceedings of Symposia in Pure Mathematics* Vol. **55.1**: *Motives*. See [here][1] for the AMS ebook collection version (you might be able to access this from within your institution). See in particular **Proposition 2.6** of Milne's chapter *Motives over finite fields*.

**Remark.** As I understand it, the same is not expected to hold over basically any other type of field. For example, it should be easy to write down a Hodge structure that is not in the full abelian tensor subcategory generated by the Hodge structures of weight $1$.


  [1]: http://www.ams.org/books/pspum/055.1/