Some reasons for using an integrated skills approach are…
That the use of language for meaningful purposes requires the use of integrated language skills.
And, that each skill reinforces the other skills.
In addition...
Using “real” or “natural” language production and reception in classes is a good match for the social interactions that occur in communicative activities; and,
The more interesting the activities are, the more likely they are to motivate the students, and the more likely they are to produce greater language retention.