In addition to the above answers involving spinors and/or octonions, you might be interested in Cartan's original construction of the triality automorphisms, which is very explicit and takes just a couple of pages in his beautiful little paper Le principe de dualité et la théorie des groups simples et semi-simples (Bull. Sc. Math 49 (1925), 361–374). The description is in Section 5 of that article, mainly on pages 368 and 369, though you'll probably be interested in the very geometric construction that he makes to interpret the outer automorphisms in the conceding sections 6 and 7 of that paper.

The main difference from what has been said in answer so far is that, instead of constructing spinors (which, of course, he already knew how to do at that point), Cartan works on the centerless simple group $G = \mathrm{SO}(8)/\lbrace\pm\mathrm{I}\rbrace$ (which he calls the 'adjoint group' of $\mathrm{SO}(8)$), since the triality automorphisms actually do act on $G$ and not just on its Lie algebra. He then uses a clever choice of notation to write down an explicit action of $S_3$, the symmetric group on $3$ letters, on the Lie algebra of $G$ in such a way that these automorphisms are Lie algebra automorphisms that preserve the obvious maximal torus and yet are outer because they don't come from the Weyl group.

Another great source, of course, is Chevalley's beautiful little book The algebraic theory of spinors (1954), the last chapter of which is all about triality. He takes pains to do everything over general fields as well, so it's quite a useful treatment.

In addition to the above answers involving spinors and/or octonions, you might be interested in Cartan's original construction of the triality automorphisms, which is very explicit and takes just a couple of pages in his beautiful little paper Le principe de dualité et la théorie des groups simples et semi-simples (Bull. Sc. Math 49 (1925), 361–374). The description is in Section 5 of that article, mainly on pages 368 and 369, though you'll probably be interested in the very geometric construction that he makes to interpret the outer automorphisms in the concluding sections 6 and 7 of that paper.

The main difference from what has been said in answer so far is that, instead of constructing spinors (which, of course, he already knew how to do at that point), Cartan works on the centerless simple group $G = \mathrm{SO}(8)/\lbrace\pm\mathrm{I}\rbrace$ (which he calls the 'adjoint group' of $\mathrm{SO}(8)$), since the triality automorphisms actually do act on $G$ and not just on its Lie algebra. He then uses a clever choice of notation to write down an explicit action of $S_3$, the symmetric group on $3$ letters, on the Lie algebra of $G$ in such a way that these automorphisms are Lie algebra automorphisms that preserve the obvious maximal torus and yet are outer because they don't come from the Weyl group.

Another great source, of course, is Chevalley's beautiful little book The algebraic theory of spinors (1954), the last chapter of which is all about triality. He takes pains to do everything over general fields as well, so it's quite a useful treatment.

Finally, Cartan also wrote about the tie of triality with $\mathrm{F}_4$ acting irreducibly on the $26$-dimensional space of traceless $3$-by-$3$ Hermitian octonian matrices in Section V of his paper Sur des familles remarquables d'hypersurfaces isoparamétriques dans les espaces sphériques (Math. Zeitschrift 45 (1939), 335–367). The relevant passage is on pages 354 and 355, where he explains the construction.