I case it helps, this is how it was taught to me when introducing proofs.

For your example case, it's simply the matter of seeing multiplication as repeated addition. So, 2x is x+x, and from this follows:

2(x+y) = (x+y)+(x+y) = x+y+x+y = x+x+y+y = 2x+2y

However, for the other case, when we convert the multiplication by 2 to an addition and back:

2(xy) = (xy)+(xy) = xy+xy = 2xy

Now we can show that this works for n instead of two:

n(x+y) = (x+y)+..[n times]..+(x+y) = x+..[n times]..+x + y+..[n times]..+y = nx+ny

And for multiplication:

n(xy) = xy+..[n times]..+xy = nxy