[impendia]

I'm a math researcher, and I'll explain why I like these sorts of definitions.

In the first place, what you quoted is not a formal, precise definition; it is not a substitute for such a definition, nor is it intended to be one. The Wikipedia page you mention has a precise definition further down the page.

So what, then, is the purpose of the description you quoted? Why include it at all?

Because it's how mathematicians conceptualize of what a field is. It is the peg we hang our hat on; it is what we remember. A mathematician who has seen fields would be able to fill in the details; and if not, they would know to look up the precise definition in a textbook.

In short, these definitions are how we keep track of the forest at the same time as the trees.

I should note that taste differs among mathematicians, and you can find different styles of exposition in math books. Some are very formal and precise; whereas others are more informal and have lots of handwavy statements along the lines of the one you quoted.

[jhanschoo]

I'd also like to point out that it is also frequent that one encounters equivalent but different formal definitions for the same mathematical structures, and this is why the informal descriptions are important as well.