Suppose the red subgraph is not strongly connected, i.e., there are two vertices $a,b$ such that there is no red path from $a$ to $b$. Then in paricular the edge $a\to b$ is blue. For any other vertex, at least one of $a\to c$, $c\to b$ is blue, hence $c$ is weakly connected to $a$ via blue edges: $c\to b\leftarrow a$ or $c\leftarrow a$. We conclude that the blue subgraph is weakly connected.
If we ask for strongly instead of weakly connected, this need not be the case: In $G$, pick any two vertices $a,b$, colour each edge staring in $a$ and/or ending in $b$ red and each edge starting in $b$ and/or ending in $a$ blue and colour all other edges arbitrarily. Then the red subgraph is not strongly connected because there is no red path $b\to\ldots \to a$, and the blue subgraph is not strongly connected because there is no blue path $a\to\ldots\to b$.