The double struck symbol or letter Z is used to indicate integers because it comes from the German word "Zahl," which means "number." It was first used as such in a book entitled "Elements of Mathematics: Algebra" first published in 1939.
The book was written by a French collective of mathematicians known by the pseudonym N. Bourbaki or Nicolas Bourbaki. Their work became known as the standard reference in the field of mathematics and their use of the symbol N became an accepted method of denoting integers. A double struck capital letter I is sometimes also used to denote the ring of integers.