Irony is created when what we expect is different from what actually occurs. Verbal irony, specifically, occurs when one says that opposite (or something different from) what is true. For example, if you break your leg in three places and someone asks you if it hurts, and you say, "Yeah, just a little," you'd be employing verbal irony. Your breaks would hurt much more than "just a little"--they'd hurt a lot--and so you'd be saying the opposite of what is true. Verbal irony often employs sarcasm, sort of poking fun, as you can see from this example.
The narrator, perhaps Austen herself, is employing irony in this opening line because the idea that every wealthy bachelor must be looking for a wife is most certainly not a universal truth that everyone believes. It is something that a person like Mrs. Bennet believes, and she immediately shows herself to be self-centered and gossipy and ridiculous. She is so silly, in fact, that we quickly identify her as a character with whom we are not supposed to agree. Much more reasonable and level-headed characters like Mr. Bennet or Elizabeth Bennet do not subscribe to this idea. Therefore, we can ascertain that the narrator is not serious in making such a statement but rather is poking fun at it, using verbal irony to do so.