Characteristics of an isogeny between super-singular elliptic curves
Given two elliptic curves $E_1$ and $E_2$, an isogeny from $E_1$ to $E_2$ is an algebraic map $\phi$ (it is basically defined from rational functions) that sends the neutral element from $E_1$ to the neutral element of $E_2$.
As stated in Theorem 4.8 in page 71 of The Arithmetic of Elliptic Curves by J.H. Silverman:

Let $$ \phi : E_1 \longrightarrow E_2 $$ be an isogeny. Then $$ \phi(P+Q) = \phi(P) + \phi(Q)\quad \text{for all } P, Q \in E_1.$$

In other words, the isogeny is a group homomorphism (the group law of the curve $E_1$ is preserved after the transfer to the curve $E_2$).
The proof involves the divisor class groups of the curves, using the fact that the isogeny induces a group homomorphism between them.