Why is the alkene a more stable species in this case?
I don’t think that your interpretation is what the author meant. I think that they are comparing the reactivity (or inverse stability) of the alkene to that of the carbocation. I think we can both agree that a carbocation is a much more reactive species than an alkene.

When it comes to comparing the products of a nucleophilic substitution to those of an elimination, I don’t think there’s an easy call that can be made. It depends far too much on variable factors such as the steric congestion around the reactivity site, the substitution pattern of the double bond, the size and shape of the nucleophile etc.

In the case of this specific reaction, however, we can make that call. The only mildly nucleophilic species is the hydrogen sulphate anion $\ce{HSO4-}$ deriving from deprotonated sulphuric acid. Hydrogen sulphate has three equivalent oxygens that each carry one third of the negative charge, and they are also practically symmetry-equivalent (not ideally, though, since the proton will angle off). The high oxidation state of the central sulphur atom reduces the electron density on the oxygens. Each of these oxygens is therefore a very bad nucleophile — compare to typical sulphonic acid residues such as tosylate or mesylate which are equally terrible nucleophiles. This should make a nucleophilic attack highly unlikely.

A second thing to remember is that sigma bond does not equal sigma bond. Usually when discussing sigma bonds, $\ce{C-C}$ or $\ce{C-H}$ bonds are meant which are very unpolar, have a bond dissociation energy of something around $\pu{400 kJ/mol}$ and are thus very hard to dissociate. Here, we would be making a $\ce{C-O}$ bond. But not any kind of $\ce{C-O}$ bond: the oxygen belongs to a sulphate residue. This makes the $\ce{C-O}$ bond very weak and very easily cleaved, especially since sulphate is a very good leaving group.

Therefore, in this specific case we can actually make the call and say that the elimination product is more stable than a product of a nucleophilic substitution.

One more thing: you do write cleavage of a sigma bond. But what is actually happening is akin to a deprotonation: a proton leaves leaving its electrons behind in a type of acid-base reaction. As you may know, these are typically very rapid as protons have an easy time ‘jumping’ from one molecule to another if the overall conditions for acid-base reactions are favourable.