How is (2,3) in the span of u=(1,3) and v=(2,1)?"
This is asking "how can we get to (2,3) using u and v.
Well, u is pretty close to (2,3), we just need to be able to move one in the x direction, that is, we need to add (1,0) to (1,2) to get to (2,3). Can we do this with u and v?
Well the diagram shows how we can go 5 units in the x direction, since (5,0)=3v-u.
So we just go a fifth of this: (1,0)=(1/5)(3v-u)=(1/5)3v-(1/5)u= (3/5)v-(1/5)u
So we can get to (2,3):
This is the same answer as what you'll get by solving the augemented matrix, but this is more geometric.