Given $$ A \to g A g^{-1} - i dg g^{-1} $$ where the gauge field one-form $A$ is $A = A_{\alpha \mu} t^\alpha dx^\mu$, thus, $$ dA\\ \to d(g A g^{-1} - i dg g^{-1}) \\ = dg \wedge A g^{-1} + g dA g^{-1} - g A \wedge dg^{-1} + i dg \wedge dg^{-1} \\ = g dA g^{-1} + (dg \wedge A g^{-1} - g A \wedge dg^{-1} + i dg \wedge dg^{-1}) $$ where $d$ is exterior derivative.
- For Abelian case, the part in $(\dots)$ drops out, therefore $F=dA$ is covariant as in Electromagnetism.
- For non-Abelian case, the part in $(\dots)$ is non-zero, therefore you need the extra $A\wedge A$ as in the non-Abelian $F = dA + A\wedge A$ to make $F$ covariant.