The sum of two sets of numbers $S,T$ is defined as: $$ S+T=\big\{z \mid \exists x \in S \quad\exists y \in T \text{ such that } x+y = z\big\} $$
I'm not really sure what $S+T$ is. If $S=\{s_1,s_2,\ldots,s_n\}$ and $T = \{ t_1, t_2, \ldots,t_n\}$ then is $S+T=\{s_1+t_1,s_2+t_2,\ldots,s_n+t_n\}$? Is $z=S+T$ or $S+T=\{z_1,\ldots,z_n\}$?
The notation is obfuscating. Another way to write the same thing is
$$S+T = \{x+y\;|\;x\in S, y\in T\}$$
It's just the set of all possible sums resulting from adding an element of $S$ to an element of $T$.
Sometimes it's easier to see visually that it's the same as $$\bigcup_{s\in S}(s+T)$$
The sets $s+T$ are copies of $T$ that have been shifted by $s$. Then just combine all of these shifted copies.
I'm assuming here you are working with subsets of $\mathbb R^n$.