This phrase was born from the original phrase 'fifth wheel' and throughout time the number of wheels was reduced to three - ha! How it came to mean what it does today - a third person attending a date clearly meant for just two people - is anyone's guess. ( source )

There are multiple ideas about the origins of this phrase, they range from religious to capital punishment. Read about the many ideas here .  When it is used today it is generally meant to indicate that if you try a third time you will be successful.