Full faith and credit refers to the phrase in Article IV, Section 1 of the Constitution which states, "Full faith and credit shall be given in each State to the public acts, records and judicial proceedings of every other state." In other words, full faith and credit means that every state in America will respect and uphold the rulings of the courts in other states.
Full faith and credit is basically just a reference to the clause in the Constitution which mandates that courts support, as opposed to undermine, each other in different states. If the courts did not respect the rulings of courts in a different state, then this could lead to unnecessary conflicts. Also, convicted persons could simply flee to other states to try to get their rulings overturned. So, the full faith and credit clause allows the legal system to function more efficiently and with fewer conflicts.