Does MST exclude the edge belongs to the cycle of the graph?
I think you need to be more precise in your reading of the statement, though it could be better phrased. A different version is "if $e$ is an edge of $G$ and it is not in any cycle in $G$, it must be in any MST of $G$". This doesn't say that edges that are in cycles can't be in the MST, it just says that any edge that isn't in a cycle must be.

To see why this is true, imagine an edge that is not part of a cycle; it has two endpoints, both of which must be part of the MST (otherwise it wouldn't be spanning). So how do we get to those endpoints? If we don't use that edge, we can't reach one of the endpoints (if we could reach both, then the edge would be part of a cycle), thus that edge must be in the MST.