It's generally easier to deal with fractions where the numerator is complicated; for instance, $\frac{\sqrt{3}-1}{2}=\frac{\sqrt{3}}{2}-\frac{1}{2}$, but there's no similar obvious way to split up $\frac{1}{\sqrt{3}+1}$.
As a result, it's often more useful to rationalize. Before calculators, when it was hard to check if two expressions represented the same number, it was common to always rationalize. These days it's less common to insist on it, but being able to do it when needed is important: situations like that come up a lot in calculus, and rationalizing at the right time can make a problem much simpler, or even be the difference between a problem looking unsolvable, and turning it into something easy. (Of course, in calculus the fractions usually have variables, so it's harder for a calculator to check if they're equal.)