Geographically, the tropical region is a wide swath around Earth's middle stretching from the Tropic of Cancer, just south of Miami, to the Tropic of Capricorn, which cuts Australia almost in half. It's about one-quarter of the globe and generally thought of as hot, steamy and damp, but it also has areas of brutal desert.
In fact, the tropics make up about two-fifths of the globe. To be more precise, I mean that about two-fifths of the total area of the surface of the earth is between the two tropics. The tropics are at latitudes +23.5 and -23.5 degrees (I'll use + and - for north and south here.) So it's easy to see where the "one-quarter" figure might come from -- the tropics span a total of 47 degrees of latitude, out of the full range of 180, and 47/180 is essentially 1/4.
But there's an interesting fact about the surface area of a sphere. Take a sphere of radius r. Cut it with two parallel planes of distance h apart. Then the area of your slice will be h/(2r) times the surface area of the sphere, or 2πrh, regardless of the way the sphere is cut; notice that this is also the surface area of a cylinder of height h and radius r. I saw this demonstrated once, when I first saw this fact in a calculus class, by cutting a spherical loaf of bread into slices of equal thickness; the slices get varying amounts of the interior of the bread but all get the same amount of crust.
The thickness of the slice in question, for the tropics, is (2 sin(23.5o))r, where r is the radius of the earth. Thus this slice makes up sin(23.5o) = 0.398 of the earth, which is just under two-fifths. I knew from the moment I read this fact that one-quarter was an underestimate, but I suspected that perhaps it was more accurate than, say, "one-third". But it's not.
(The actual point of the article is not that the tropics make up one-fourth, or two-fifths, or whatever fraction of the earth, but that they are "widening"; this uses an atmospheric definition of the tropics which is different from the astronomical one implied by the quote above.)