Movements of the tectonic plates - collison between Indian Plate & Eurasian Plate
As the continental crust is too buoyant (light) to be subducted, when two continents collide they smash against each other creating a mountain range, in this case the Himalayas.
This process happen trough thrusting and folding of the plates, leading to something called lithospheric shortening. Which basically means that the plates (lithosphere is the stuff of which plates are made of) gets shorter. In other words: the area of the colliding continents gets smaller. And the missing area is used to build up the mountains by thrusting and folding, a process that is nicely illustrated by the following animated GIF that I put together from a video by the Department of Earth Sciences of UC Santa Barbara.



In this question, this process is also discussed.

In regard  of the Monsoon, it is not a direct consequence of this collision. But it is due to the fact that Eurasia is a big continent close to the equatorial Indian ocean. Now, as this collision helped to put together Eurasia and bring it closer to the equatorial Indian ocean in the area associated with the Monsoon, you can argue that played a significant role in creating the Monsoons as we know them.