A few complements to Geoff's answer and Agol's comment:

1) Dickson's theorem is worked out in detail (somewhat lengthy) in Gorenstein's
Chapter 2, page 44+, as Theorem 8.4. His section 8 is devoted to a study of this matrix group and its projective quotient. The treatment is elementary but not conceptual. In the exceptional case of the field of 9 elements, the proper subgroup you might generate is of order 120 and has the simple group of order 60 as a quotient.

2) Larger matrix groups are definitely more complicated, though the theme of generation by unipotent matrices (geometrically, transvections) is in any case prominent in the study of such classical groups. For instance, when you look at $3 \times 3$ matrices over a finite field, you easily find two unipotent Jordan blocks (with a single block just above the diagonal) which fail to commute and only manage to generate a subgroup of upper triangular matrices.

3) As Agol comments, you still need to get your arbitrary non-commuting unipotent matrices to be conjugate to a pair treated in Dickson's theorem. What Agol suggests is in the more sophisticated but also more conceptual framework of Chevalley groups (as in Steinberg's old Yale lectures), or more generally semisimple linear algebraic groups. Looking at $\mathrm{SL}_2$ in this spirit, there are two Borel subgroups containing a typical maximal torus such as the group of diagonal matrices (and any two distinct Borels intersect in such a torus). In turn, the unipotent subgroups (say upper and lower unitriangular matrices) suffice to generate the entire group. Such a unipotent group is cyclic over the prime field, but otherwise elementary abelian (which creates complications for applying Dickson's theorem). Moreover, the quotient of the full group by a Borel subgroup is a projective line, on which the torus has precisely two fixed points, etc. This is where Agol's comment comes from.

4) Most of this Chevalley group formalism is well encoded in the more elementary theory of split BN-pairs, where it's still fairly easy to see conceptually how the generation by unipotents works. See Steinberg's lecture notes (now online at his UCLA homepage) or Carter's 1968 book for results in this direction.