$H_0^1$ is functions of $H^1$ that have a property of vanishing (i.e. go to zero) on the boundary of the function's domain.
So $$H_0^1 \subset H^1$$
However, if one considers the sum:
$$v_1 + v_2$$
where $v_1 \in H^1$ and $v_2 \in H_0^1$, then what can one say about the set where this sum is?
I interpret that one cannot say $v_1+v_2 \in H^1$, because there could be more sets in $H^1$. Thus it's possible that the sum would (term-wise) be in $(H^1 -H_0^1) + H_0^1$. Or does this mean the same as in $H^1$?
From $H_0^1\subset H^1$ and $v_2\in H_0^1$ it follows that $v_2\in H^1$.
Because $H^1$ is a vector space, it means that the sum of elements in $H^1$ is again in $H^1$. This means, that $v_1+v_2\in H^1$ follows from $v_1\in H^1$ and $v_2\in H^1$.