Most of the naturally occurring elements occurs in the form of isotopes and if different isotopes present in the element are significant in quantity then by calculating their weighted average we get fractional atomic weight.

Note that in question it is given that most of the element's atomic weight is fractional not all elements. Because for example atomic weight of Cl = 35.5 which is fractional but atomic weight of C = 12 which is integer. That is why we can't say all element's atomic weight is fractional. The reason is explained below $$\to$$

Naturally available two isotopes of Cl are 75% $${}_{17}C{l^{35}}$$ and 25% $${}_{17}C{l^{37}}$$ . Here both isotopes are significantly present inside Cl. So their weighted average depends on both the isotopes. And by calculating we find atomic weight of Cl = 35.5 (It is a fraction). There are 3 main naturally available isotopes of Carbon (1) 98.9% $${}_6{C^{12}}$$ (2) 1.1% $${}_6{C^{13}}$$ and (3) 0.0001% $${}_6{C^{14}}$$ . As $${}_6{C^{13}}$$ and $${}_6{C^{14}}$$ are negligible compared to $${}_6{C^{12}}$$ . So their weighted average depends on only $${}_6{C^{12}}$$ . By calculating we find atomic weight of C = 12.0107 $$\simeq$$ 12 (integer)