The British Empire was a large group of countries and territories that were controlled by the British Government. A 'demographic' is just another word to describe a group of people, so the 'demographics of the British Empire' means the people who lived within it.
When the British Empire was at its height, most of the countries it controlled were in what we now call India and Africa. There were also many colonies in North America and the Caribbean. People from all of these places were part of the Empire. However, most of the people living in the British Empire were from Britain (England, Scotland, Wales, and Northern Ireland). Many people from India and other parts of the world also travelled to Britain during the time of the Empire, so the people living in Britain also included people from many different backgrounds and cultures.