He's probably talking about the top officers, not the entire civil service.

The Indian Civil Service had two divisions: covenanted and uncovenanted. The covenanted division retained all the top posts and consisted almost entirely of white British men. Beginning in the early 1900's they let in a few token Indians. This accelerated after the Government of India Act of 1919, but even at independence in 1947 about 70% of covenanted officers were British.

But of course 1000 officers (actually, there were 1010 to be precise) could not govern a country of over 300 million people. Day to day administration was actually done by the uncovenanted part of the Civil Service, which was almost exclusively Indian. The uncovenanted Civil Service numbered around 130,000 civil servants at the time of independence.

So in that sense it's no different from asking "how could one man rule the Mongol Empire", or "how can one President govern 350 million Americans"? Obviously, through a lot of delegation.

It's useful to think of the administrative hierarchy as a pyramid. At the top, you had the British king and parliament. They appointed a viceroy for India with his staff. He presided over the Indian Civil Service with an upper tier of 1000 covenanted officers, who were almost exclusively British, and a lower tier of 130,000 unconvenanted officers who were almost exclusively Indian.

All of this was mediated by the police and military, who are not counted in these totals. The Indian police forces numbered over a million, and they were almost exclusively Indians except for a handful of British officers. The military grew dramatically over the course of WW2, and numbered around 2.5 million towards the end. Again, it was overwhelmingly Indian with a sprinkling of British officers.

So in short, a thousand civil servants were able to rule India because they had the cooperation of millions of Indians who followed and implemented their orders.

This, also, was the point of attack of Gandhi's Civil Disobedience campaign. He understood that British rule was a very thin facade that could be torn down if Indians simply refused to accept orders.

Edit: a good source for this is Clive Dewey's book Anglo-Indian Attitudes: the Mind of the Indian Civil Service.

A lot of people find it somewhat amazing that 'tiny' Britain could rule such a vast India. What we need to understand about the British Indian Empire, however, is that it was a piecemeal construction, built over nearly a century.

Beginning from the first British territory in Bengal, in 1757, the British Conquest of India was completed in 1845-7, after the Anglo-Sikh Wars.

In Bengal, the East India Company inherited an already well constructed bureaucratic system, built specifically for 'foreign' control of the local population. The rulers of Bengal before the British were Islamic people of Central Asian origin. As the other commentator describes in his brilliant answer, bureaucratic systems are constructed like a pyramidical hierarchy. When the British conquered Bengal, they only replaced the top tier of the already existing hierarchy.

The EIC also conceived of a geopolitical policy called the 'subsidiary alliance system'. In this, they would not displace the local king after defeating him in war, but make him a partner, he continued as king, with British support - though almost always, it was only in name.

So, again, the hierarchy did not have to be constructed from bottom up. All that was needed was a takeover at the top. Actually, one company buying another is not a bad example. The EIC was, literally, a company!

Again continuing from the other comment, what Gandhi did was, he delegitimised this old ruling class, whom the British relied on to maintain their legitimacy. Once a new middle class leadership became more representative of the 'nation', emerging as it was, the old system literally withered away.

The British government did try to maintain legitimacy by allowing limited representative government, directly under the Crown, but that was a concession similar to Glasnost and Perestroika. It only called into further question their 'hypocrisy' in preaching about freedom, while running a brutal regime.

This shows that the British could not run India without local partners, who had some kind of popular legitimacy.

I'm afraid Ferguson is not a very good historian.

(For early years of how the EIC gained power in India, I recommend John Keay's The Honourable Company. JL Nehru's Autobiography, written in the 1930's is a good account of these power shifts, by an insider. He became India's first Prime Minister.)

Edit: one spelling.