The fact that 1 + 1 = 2 is not an axiom of the Peano arithmetic, but it is fairly easy to demonstrate. The Peano axioms for natural integers are all based on the notion of "successor". The only named integer is zero (0), and then there are five axioms revolving around zero and successors of natural integers.
The Peano arithmetic introduces further axioms for various operations on integers. It defines two axioms for the addition. For any natural integers a and b, and S the notation for the successor of a given integer:
(1) a + 0 = a (2) a + S(b) = S(a + b)
Let 1 = S(0) and 2 = S(1). Then:
1 + 1 = 1 + S(0) (definition of 1) = S(1 + 0) (axiom 2) = S(1) (axiom 1) = 2 (definition of 2)
The intuition behind the mechanics of axiom 2 is that adding c to a means taking c times the successor of a (that is,
S(S(S(S(...(a))))) with c S marks). In more informal terms,
a + S(b) = S(a + b) should be thought as
a + c = S(a + (c-1)). The right operand is "decreased" by prefixing the whole thing with the successor operator. Eventually it will reach zero and the number of time we needed to decrease it to reach zero is kept track by the number of time we marked the S notation.