Role of xor in ARX construction
Xor is the "addition" operator for algebra with boolean operators, while AND is the "multiplication" operator for boolean operators. "Regular" addition and multiplication are the addition/multiplication operators for integers. So speaking of "linearity" in the sense of algebraic compatibility, XOR and integer addition are non-linear. I feel like this question/answer(s) sums it up pretty well. 

As for why xor is required/why AR is not sufficient, technically, AR is equivalent to ARX, but less efficient. See the paper Rotational Cryptanalysis of ARX . There are some details to it:


  We also show that the
  AR systems
  , that do not use XOR, are theoretically
  equivalent to ARX systems. However, we prove that they are less secure with
  the same number of operations, because of the linear mod 2**n approximation.
  It is also easy to prove that omitting addition or rotation is devastating, and
  such systems (XR and AX) can always be broken.