Why did Russia begin to pressure Great Britain in inner Asia?

Answer 1

The 19th century rush for power and influence in Central Asia (especially Afghanistan) and India is called 'the Great Game'. Russians then as now wanted a level of control over the areas immediately bordering them and they saw Britain's invasion of Afghanistan as an immediate strategic threat to themselves. The same happened when later on the British invaded and occupied Tibet.

So it was not so much "Russia pressuring Great Britain" as Russia counterpressuring Britain's expansionist policies in the area. While they were at it, Russia was not above conquering and annexing several large areas in Central Asia an northern China for themselves, whether they were threatened by the British or not.

Answer 2

In about 1850, Russia began to pressure different parts of the British Empire. Since Russia was land locked in the Winter, it pushed for some type of warm water port in British held colonies and even Turkey and then Persia. India was a possibility as an opening to the Indian ocean. Turkey and Persia both had warm weather ports in the coldest Winters.