So having two groups $H$ and $G$ the direct product $H \times G$ is the set of ordered pairs $(h,g)$, where $h\in H$ and $g\in G$, and multiplication is carried out componentwise, i.e. $(h_1,g_1)(h_2,g_2)=(h_1 h_2,g_1g_2)$, such that the direct product is again a group.
As for Matrix Lie groups, the elements can be represented by matrices which act on vectors for example. How is this action on vectors carried out in the direct product? Take as an example the direct product of $SU(2) \times U(1)$ (which is not quite, but almost isomorphic to $U(2)$). Using the defintion I would write down pairs of matrices $(S,U), S\in SU(2), U \in U(1)$. But how do these pairs act on vectors now? And vectors of which dimension? If both groups are represented in the fundamental representation, do these pairs act on three dimensional vectors $(s_1,s_2,u)^t$?