In the clause the term "jurisdiction" was meant as Ties/Allegiances/Loyalties.
I am having trouble finding the website that had some speeches and debates from the men who authored the clause. I am at work now, I think I have it bookmarked at home ill check when I get there.
That may have been what some who debated the issue thought it meant, but the word has a very specific meaning. It literally means "administration of justice." The government has the power to administer justice. Individuals have ties/allegiances/loyalties.
The point of the phrase was to recognize that the government could "administer justice" to former slaves (had jurisdiction over them), even though they were not citizens, and therefore they ought to be.