How constant value 0x63 in affine transformation is chosen or generated while AES Substitution Box is formed?
According to the Rijndael AES proposal (emphasis mine):


  We  have chosen an affine mapping that has a very simple description per se, but a complicated algebraic expression if combined with the ‘inverse’  mapping. It can be seen as modular polynomial multiplication followed by an addition: $$b(x) = (x^7 + x^6 + x^2 + x) + a(x)(x^7 + x^6 + x^5 + x^4 + 1) \mod x^8 + 1$$ The modulus has been chosen as the simplest modulus possible. The multiplication polynomial has been chosen from the set of polynomials coprime to the modulus as the one with the simplest description. The constant has been chosen in such a way that that the S-box has no
  fixed points (S-box($a$) = $a$) and no ’opposite fixed points' (S-box($a$) = $\bar a$).


I'm not personally aware of any other design considerations in the choice of the additive constant 0x63, although it's worth noting that it does have a fairly nice and simple binary representation (0x63 = 0b0110&thinsp;0011), which might perhaps simplify some implementations slightly.