Yeah, we said it: Wells' writing style is realistic. How about the whole invisible man thing? you might ask. And the fact that Griffin just happens to run into an old college acquaintance by accident? Well hey, we didn't say the plot was realistic, just the writing style. There's a big difference.
The mariner at Port Stowe tells us what we should expect from realistic prose: "Names and everything" (14.29). That is, the mariner believes what he reads in the newspaper about the Invisible Man because the story has a lot of believable details, like, say, names.
Now, check out the first chapter again, when the Invisible Man is at the Coach and Horses: "He wore a dark-brown velvet jacket with a high, black, linen-lined collar turned up about his neck" (1.16). Look at those details – the collar is linen-lined, the jacket is velvet. So this must be a true story – after all, if the narrator was just making this story up, would he include those details? Well, yes: Wells is making this whole thing up (brilliantly, by the way) and he includes these details to make the story seem more realistic. Booya.
Wells was actually famous for taking an impossible situation (time travel, alien invasion, invisible man, you name it) and thinking through the realistic issues involved. For example, what physical issues might the IM have? Well, his clothes are still visible, but when he takes them off, he has some trouble walking down the stairs (21.1). Also, and this is a bummer, he can't blink because his eyelids are transparent (20.45).
Joseph Conrad (who wrote Heart of Darkness) wrote Wells a fan letter about The Invisible Man, where Conrad called him "the Realist of the Fantastic." We can see why. He takes an impossible scenario, thinks through some of the logical issues around it, adds details, and allows characters to have realistic reactions to impossible events.