The Citizens of each State shall be entitled to all Privileges and Immunities of Citizens in the several States. A Person charged in any State with Treason, Felony, or other Crime, who shall flee from Justice, and be found in another State, shall on Demand of the executive Authority of the State from which he fled, be delivered up, to be removed to the State having Jurisdiction of the Crime.
This means that if I live in New Hampshire, I am still allowed to own property or work in Vermont. I can’t be prevented from traveling to another state. If I decide to kill someone in Florida and then go to New Mexico, I can be sent back to Florida to stand trial for that murder.
No Person held to Service or Labour in one State, under the Laws thereof, escaping into another, shall, in Consequence of any Law or Regulation therein, be discharged from such Service or Labour, but shall be delivered up on Claim of the Party to whom such Service or Labour may be due.
The Thirteenth Amendment got rid of this one. Basically, an enslaved person who escaped could be returned, even if they made it to a state that did not permit slavery. When slavery was abolished, this clause became irrelevant.