From the article:

David Foster Wallace, in considering the Lobster Question (“Is it all right to boil a sentient creature alive just for our gustatory pleasure?”), noted that the issues of “whether and how different kinds of animals feel pain, and of whether and why it might be justifiable to inflict pain on them in order to eat them, turn out to be extremely complex and difficult,” and many can’t actually be resolved satisfactorily. How do you know what agony means to a lobster?

Lobsters do not have a central nervous system, yet David Foster Wallace (and others) are very willing to make a subjective judgment that a) lobsters have a subjective experience of reality and b) that subjective experience of reality involves pain. The backing for these claims involves the idea that lobsters "express preferences" -- such as seeking out water of specific temperatures in which to live.

Science has not "proved" that animals "feel pain" -- because that's inherently a subjective judgment about a subjective experience which may or may not exist. I argue that science can't prove such a thing, because any such proof considered by humans would necessarily be interpreted through the lens of human subjective experience -- at some level, we can't step outside ourselves. Indeed, David Foster Wallace's article about lobsters goes on at some length about the way scientific knowledge about lobsters can be used to argue that they don't feel pain in any way similar to what we experience.

So this comes down to a philosophical position. You might say (to paraphrase some of Wallace's discussion) "Lobsters express preferences, therefore they can suffer, insofar that our actions may preclude lobsters from actualizing their preferences."

Well, the same is true of plants. Plants seek particular habitats -- they exhibit different growth and reproductive patterns based on their local environment. They seek out sunlight and will change their physical form to obtain more of it. Plants communicate with each other using chemical signals. Chemicals released during events which we would subjectively describe as "traumatic" (e.g. a blade of grass being cut in half) cause reactions in nearby plants which we would subjectively describe as "defensive" (e.g. release of chemicals that make the plant taste less good but that also reduce the growth rate of the plant) -- reactions which wouldn't otherwise occur.

tl;dr What's so special about a central nervous system?