The Local Group is only one of many, many clusters of galaxies. All of the galaxies are all moving away from each other causing more and more space to come between them. This means the universe is expanding. That discovery is what led to

Answer: this theory/hypothesis is a result of another idea, i. e. idea according to which after the Big Bang (hypothesis, again) and since then cosmos/Universe is expanding so all the object found in the universe are moving from each other. There is another idea that after expanding will follow shrinking. We found ourselves in pure theorizing, of course.