Originally zero was not considered a natural number. In fact it is one of the most unnatural numbers. Since the word "progression naturelle" was introduced by Chuquet, in 1484 and "natural number" by Emmerson in 1763 always the sequence starting with 1 has been addressed. "Natural number, defined as the numbers 1, 2, 3, 4, 5, etc., appears in the 1771 Encyclopaedia Britannica in the Logarithm article." http://jeff560.tripod.com/n.html Cantor for instance (although rarely using this wording but prefering "integers" (ganze Zahlen) mentiones the "natural sequence of numbers" (natürliche Zahlenreihe): $1, 2, 3, ...,\nu , ...$" in Über unendliche lineare Punktmannigfaltigkeiten Math. Annalen 15, (1879). Also Peano started with 1 in his first papers. Later he switched to 0 like many others. (See the answer by Mauro Allegranza)
Where did these conventions come from? What was the reason for this change? Cantor explained it in a letter to Dedekind of Juli 28,1899:
The sequence of ordinal numbers $0, 1, 2, 3, ... ,\omega_0, \omega_0 + 1, ..., \gamma,.. $ has the property that every number is the ordinal type of the preceding numbers. Without including zero this property would not be satisfied for the finite part.
I think that this is the reason.