I am not sure exactly how they are doing it, but there is certainly no random generator at work here. I have carried out an experiment - using the first chapter of some books I have on my phone and from literature.org the following matches were made:

Nicolas Nickleby by Charles Dickens, "I write like Charles Dickens".

Impressive, thinks I. Lets try:

Emma by Jane Austin, " I write like Charles Dickens". Well, you know, there actually is some truth in that in terms of sentence structure and grammar.

What about:

A Scandal in Bohemia, by Arthur Conan Doyle. But I changes the name Sherlock Homes to Jim McGill and Irene Adler to Jessica Adams, juut yo reduce the chance of some skulldiggery.

" I write like Arthur Conan Doyle"

Hmmm.
Crime and Punishment by Fyodor Dostoevsky. Used the second paragraph because the place names in the first paragraph are in a distinctive structure. " I write like Charles Dickens". Not as clear as with the Austin, buyt still, probably not a bad comparison.

Alice in Wonderland by Lewis Carroll. Name Alice changed to Jimbob.
"I write like Lewis Carroll".

And then I got bored.

Quite an impressive strike rate - I wonder what samples it uses for comparison and how many authors it has on its database, but it looks pretty good on this sample.

Oh, and this post, right up to the wors "sample" above - "I write like HP Lovecraft.