Why do disks on planes grow more quickly with radius than disks on spheres? In the book, Mr. Tompkins in Wonderland, there is written something like this:

On a sphere the area within a given radius grows more slowly with the radius than on a plane.

Could you explain this to me? I think that formulas shows something totally different:
The area of a sphere is $4 \pi r^2$, but the area of a disk on the plane is $\pi r^2$.
UPDATE:
Later in this book there is:

If, for example, you are on the north pole, the circle with the radius equal to a half meridian is the equator, and the area included is the northern hemisphere. Increase the radius twice and you will get in all the earth's surface; the area will increase only twice instead of four times if it were on a plane.

Could someone explain me this?

This is effect of double curvature. When Gauss curvature $K$ is negative (hyperbolic geometry)  it grows faster than in a plane ($K=0$) or on a sphere ($K >0 $).
IIRC (it is in the Geometry & Imagination book by Hilbert/Cohn Vossen?) the words elliptic, hyperbolic derive their name due to the way area develops in relation  to the plane.)
It becomes clearer by Gauss-Bonnet thm considering integral curvature $ \int K dA $.
However in simpler terms consider an area A spanned by constant K:
If $ A\cdot K $ is constant then $ dA/ A = - dK/K $ so hyperbolic case grows faster.
If you google for sea coral shapes with Daina Taimina's Crochet work, you appreciate that hyperbolic patches grow faster than flat ones or spheres in that order, the plane is more hyperbolic than sphere in that sense. I had to mention hyperbolic as I believe it is relevant for an answer here. 
