Some people put commas wherever they would pause. Many experts (such as the Grammar Girl, the UNC Writing Center, and Penn State's Style for Students would debunk that myth, explaining that, even though commas often go where pauses would occur, that's not reliable guidance for where to insert commas.
You have written the sentence:
Jack and Kate are leaving the province tomorrow.
in a way that is correctly written and punctuated.
So, why the extra commas? I'd guess that the writer wanted to emphasize the fact that Kate was going for some reason. If that's the case, the writer should have employed dashes:
Jack – and Kate – are leaving the province tomorrow.