Why we can add an element on both sides like this? Suppose $A$ is a set equipped with a binary operation $+$.
How I can prove if $a=b$ then $a+c=b+c$? Why we can add an element on both sides like this?

A simple explanation is that saying $a=b$ literally says that $a$ and $b$ are the exact same mathematical object. Hence 'adding' $c$ to $a$ on the right is one the same as 'adding' $c$ to $b$ on the right (adding in quotes because what I am really referring to is the given binary operation).
Note that the converse, $a+c=b+c \Rightarrow a=b$, is not guaranteed to be valid unless every element of the set $A$ has unique right inverses, and in such a case, we say that $c$ has right inverse $c_R^{-1}$ and we can apply that right inverse to both sides to obtain $a+c+c_R^{-1}=b+c+c_R^{-1}\Rightarrow a+e=b+e$ where $e$ is an identity element (required for the notion of a right inverse to make sense) and thus $a+e=b+e\Rightarrow a=b$.
