How does space expand and what does it expand into?

Some pompous twit told me that the Big Bang was not an explosion but an expansion of space. Said twit then blocked me from replying. Well upon reading that "an expansion of space" bit the first question that came to mind was..How do you expand a cubic inch? Then the two on the question line popped up.

The theory is that all the matter in the universe was originally compressed into a lump which then expanded suddenly (exploded -- bang!), with the matter and energy released continuing to spread out in all directions throughout the existing universe, a limitless empty void, so it's thought.