G1 and G2 are connected graph. As G1 $\cap$ G2 is disconnected graph, there must exist two vertices say V1 and V2 such that there is no path between them in G1 $\cap$ G2.
But as G1 is connected graph it must have some path P1 between vertices V1 and V2. Similarly, G2 must have some path P2 between V1 and V2.
Now, these P1 and P2 are not the same paths, if they are they will be present in G1 $\cap$ G2, which will contradict with our assumption.
In G1 $\cup$ G2, we have two paths P1 and P2 to between V1 and V2. That form the cycle. See the image.