Understanding the Sipser-Gacs-Lautemann theorem

Question

The class $BPP$ contains all the languages decided by a probabilistic Turing machine in polynomial time with probability of success more that 2/3 for every input.

The class $\Sigma^p_2$ contains all the languages for which there is a polinomial time Turing machine $M$ and a plynomial function $q : \mathbb{N} \rightarrow \mathbb{N}$ such that: $$ x \in L \iff \exists u \in \{0,1\}^{q(|x|)} \forall v \in \{ 0,1 \}^{q(|x|)} M(u,x,v)=1$$ Define $\Pi^p_i=\{\bar{L} : L \in \Sigma^p_2 \}$

The theorem states that the class $BPP$ is contained by the intersection of $\Sigma^p_2$ and $\Pi^p_2$.

To prove the theorem it is proved that for every set $S \subseteq \{0,1\}^m$ with $|S| \leq 2^{m-n}$ and every k vectors $u_1, \ldots, u_k$ $$\bigcup_{i=1}^k(S+u_i) \neq \{0,1\}^m$$ Where $S+u = \{ x+u : x \in S \}$ and + denotes addition modulo 2 i.e. bitwise XOR.

It is also proved that for every set $S \subseteq \{0,1\}^m$ with $|S| \geq (1-2^{-n})2^m$ and every k vectors $u_1, \ldots, u_k$ $$\bigcup_{i=1}^k(S+u_i) = \{0,1\}^m$$

I don't get why this claims imply that if a language is in $BPP$, then $$ \exists u_1, \ldots,u_k \in \{ 0,1 \}^m \forall r \in \{ 0,1 \}^m \bigvee_{i=1}^k M(x,r \oplus u_i) = 1 $$

How does the claims about sets of binary strings imply the computation above?

What I don't understand is how the translations preserve the original random strings and how can a small set of translates cover all possible random strings.