Planes through the origin are subspaces of $\Bbb{R}^3$

Remember that a plane is a copy of $\mathbb{R}^2$ which means it is not bounded. Here $u+v,ku$ lie in the same plane since there are both linear combinations of $u,v$. Recall that vector spaces are closed under addition and scalar multiplication. The fact that you want the plane to go through the origin is because subspaces of $\mathbb{R}^n$ contain the origin.

Planes are infinite in their extents. If we're given a vector $\mathbf{u}$, then the vector $k \mathbf{u}$ is obtained just by scaling the length of $\mathbf{u}$, but keeping the same direction. So, if the vector $\mathbf{u}$ lies in a plane $W$, then any vector $k \mathbf{u}$ will also lie in this same plane, no matter how large $k$ is.

The picture in your book is misleading -- it shows the plane $W$ as a bounded parallelogram, which is wrong. I guess it's a somewhat understandable mistake -- it's difficult to draw infinite things on the page of a book. But they could have drawn a region with a fuzzy border to indicate that it's infinite.