The way I thought about it: to get to the goal you have to do 20 steps: 10 times right, 10 times down. But the order of these steps does not matter, ever possible arrangement is a solution.

So for counting, you can basically think about it as a list of twenty initially empty spots. You first fill it in with your 10 down steps. The remaining 10 spots will then be the ones for the 10 right steps. So really the only choice you have to make is where to place the 10 down steps.

This question boils down to: in how many different ways can you distribute the 10 down steps over the 20 empty spots? That's 20 choose 10.