Go back to the original Senate and House debates in the 1865 congress. Also review the various States comments on the amendment. You should find that this amendment was designed specifically address the status of the newly freed southern slaves. It also would have impacted Native Americans residing in government reservations, I think.
The 13th amendment on freed the slaves, it did not yet recognize them as equal under the law. The 14th did that. That is why the "under the jurisdiction there of" was added. It didn't apply to anyone just walking into the country.