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?

For the binary operation to be well-defined, it must be the case that for any elements $a, b, c, d$ with $a=b$ and $c=d$, $a+c = b+d$. It is not at all clear in your question what further conditions there are on the binary operation $+$, but being well-defined is absolutely necessary for what you've written to make sense.
The converse is not necessarily true, however. Since you have not mentioned the existence of inverses for your binary operation, there is no reason to assume that $a+c=b+c \implies a=b$. Taking an example from ring theory, replacing your '+' with multiplication in $\mathbb{Z}_6$, we have $2 \times 3 = 4 \times 3 = 0$, but $2 \neq 4$.
See this thread for a more concise explanation of what I've said.
