Cordially means "doing something in a heartfelt way." If you cordially welcome a stranger into your house, you make him or her feel comfortable, because you show that you're truly glad for the visit.
The adverb cordially comes from the adjective cordial, which come from the Middle Latin word cordialis meaning "of or for the heart." Something that is done cordially has warmth and friendliness to it, like a host who greets people cordially at a party. In fact, formal party invitations sometimes use the phrase "you are cordially invited," which means you are happily encouraged to attend.