The expressions “right side” and “right-hand side” (and, correspondingly, “left side” and “left-hand side”) are interchangeable in most contexts. It is natural to ask, “Wouldn’t it be better to use just ‘right side’ and ‘left side’, which are shorter?”
The problem with “right side” and “left side” is their ambiguity. For example, when someone tells you:
it is not clear whether you are supposed to write your name on the correct side of the sheet of paper or in the upper right corner of the side you are looking at. By using “right-hand side” and “correct side” instead of “right side”, you can make your statements unambiguous.
“Left side” is similarly ambiguous (although its ambiguity is much less likely to cause any misunderstanding) because “left” can also be the past participle of “leave”, so “left side” could be understood as “abandoned side”.
Note that in mathematics, it is customary to refer to the sides of an equation as “the left-hand side” and “the right-hand side”. For example, when talking about$$ c^2= a^2+b^2\,, $$
you are more likely to hear that “the right-hand side is $a^2+b^2$” rather than just “the right side is $a^2+b^2$”. The expressions “right-hand side” and “left-hand side” are commonly abbreviated to “RHS” and “LHS”, respectively, in mathematical texts.