It will help if you think of it this way.

is asking how many ways can I arrange n objects in 2n spots.

So lets break this up into a two part problem. For simplicity I will use n=3 and let you generalize

So we have

So we have 3 objects and 6 six spots

_ _ _ , _ _ _

Now we ask if I put 0 ojects in the first 3 spots I can do this in

ways, but this forces me to put all 3 in the last 3 spots and I can do this in

ways

Now if I put 1 object in the first 3 spots I then have to put 2 in the last 3 I can do this in

ways respectively

Next will be 2 objects in the first 3 spots and 1 in the last three with

ways respectively

and finally 3 objects in the first 3 spots and none in the last three gives

ways respectively

So the total number of ways to do this is

Now remember that

using this we get

This is the logic you need for the general proof.

I hope this helps. Good luck.