How is it that Britain, a small European backwater, rose to conquer most of the world while larger countries (e.g. China, India) degraded?

The should be seperated into 2 different questions. The first should be "How did the west come to dominate the world? ", and the second should be "How did Britain become the greatest colonial power?" The first question is highly complex and whole areas of study have been dedicated to it. The second answer is somewhat complex, but one I can answer reasonably well.

It has to do with nations and navies. If you want to have thriving colonies, you want to have a good navy as well. Britain was not always master of the oceans, as they had contended with the Portuguese, Spanish, and french for naval supremacy. There are two battles that elevated Britain above these nations. The Spanish armada decimated the combined Spanish and Portuguese fleets, whose countries were in a union at the time. While this did not establish supremacy for England, it certainly challenged the assertion that the Portuguese and the Spanish had the greatest fleets. After this there was a couple of centuries of colonization and armed conflicts between these nations, but nothing decisive.

Then the French Revolution happens. Napoleon becomes emperor, and the napoleon wars begin. Much of mainland Europe was crushed by the French armies. But in order to crush the English, you needed a fleet to invade. The combined Spanish and french fleets were crushed at trafalgar, ensuring English domination of the sea. After the napoleonic wars, mainland Europe was devastated, while England was relatively unharmed. This allowed them to focus on empire building instead of nation rebuilding.

The other nations of Europe had severe disadvantages when compared to the four major nations I mentioned. Germany did not become a nation until 1870, at the height of British dominance. Austria was mainly landlocked, and isolated in the Mediterranean by the straight of Gibraltar. The whole of Italy was not unified until 1870. Belgium and the Netherlands had a tendency to be devastated in continental warfare, along with sweden and Denmark. The balkans were ruled by the ottomans until the 1800s. The only other major nation left out is russia, and they ate a special case. Instead of founding overseas colonies, russia simply expanded its borders parts. They did found several colonies in their new territories, which reached as far east as Western Oregon. However, these colonies had a tendency to not be profitable, as land trade routes are much more expensive than sea ropes. The main reason for this alternate strategy of empire was that for a very long time, russia lacked warm water ports. Sevastopol was finally obtained in the crimean war, but even that port had several choke points it had to go through before finally racing the Atlantic.

In summary, the reason England came out on top was open access to the Atlantic, naval domination over the French, Spanish, and Portuguese, and it's status as an island protected it from devastating continental warfare. Keep in mind that this answer is nowhere near conclusive, as there are several other factors that contributed, including religion, culture, economics, technology, and government.