Clause 1. The Citizens of each State shall be entitled to all Privileges and Immunities of Citizens in the several States.
This is pretty self-explanatory. States can't discriminate against the residents of other states.
Clause 2. 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 clause establishes the principle of extradition, requiring states that capture fugitives from justice in other states to send them back to face trial.
Clause 3. 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.
Another legacy of slavery, this is the infamous fugitive slave clause. Slaves who escaped into freedom in the North were required to be sent back to their owners in the South. The clause was nullified in 1865, when the .) The 13th Amendment banned slavery in the United States. Notice how convoluted the writing gets as the Framers of the Constitution struggle to write a fugitive slave clause without using the word "slave" ("a person held to Service or Labour") or "slavery" ("such Service or Labor may be due").