Those extremely cheap picture books they sell at places like Ollie's and dollar stores that are often not very well written and usually don't list an artist or illustrator on the cover — how do those come into existence? Who is writing them and why?

Wowww this just sent me down a RABBIT HOLE lmao.

I must say I have never thought about it - I basically never pay attention to those books as they aren't the type of thing that agents do, so I had to kind of refresh my memory and do a bit of research. So, to my understanding, the bulk of books at places like, say, Ollies, Dollar Tree, etc, probably fall into one of these buckets:

* They are remaindered books. These are regular books bought from regular publishers when they are basically clearing overstock out their warehouses -- they are sold by the pound at very steep discounts, generally to Remainder Houses, who then go on to sort and sell them at big discounts to bookstores for "bargain tables" or places like Book Outlet, or sometimes, perhaps, places like Ollies. It doesn't mean the books are out of print or unpopular, just that there are TOO MANY in the warehouse. (So if you see, say, a pile of Nicholas Sparks books or something like that at Ollies or in the Bargain section of B&N -- that's likely where they came from -- you'll know, because they are regular books from regular publishers, but they are MUCH cheaper than cover price, and they usually have some kind of mark on them, like a black ink spot on the bottom, to differentiate them from a brand-new full-price book).

* They are "mass market" books from specific companies that JUST do that kind of work, specifically for the Dollar Trees of the world. These companies do not work with agents, and frankly, I'd never heard of them, but they seem to sell a great many books to large retailers like supermarkets and dollar stores, etc. They bill themselves not as BOOK PUBLISHERS, but as makers of educational or "creative play" merchandise, including activity books and story books, often with a somewhat educational or religious (Christian) bent, or based on public domain stories like fairy tales told in a simple way, or based on licensed properties like Sesame Street.

Sidebar -- Note that I'm using the term Mass Market here a little differently than when we talk about "trade paperback" -- the Regular Largeish Paperbacks you see in every bookstore -- vs "mass market" sized paperbacks, which are those small romance novels etc that are usually on racks in the supermarket -- the concept is the same, though, the mass ones are less expensive paper, and made to be sold cheaply and in quantity.

* There are also "mass market" kids books from big publishers. However they are a LITTLE more expensive than Dollar Tree and ARE sold in regular bookstores. So, like, those floppy Dora the Explorer 8x8 books or activity books that are on spinner racks at bookstores or the airport that cost $2.99 or whatever, but DO come from publishers you have heard of. These are often (but not ALWAYS) licensed characters or based off cartoons - Dora, Doc McStuffins, Frozen, etc - or they are spin-offs of extremely popular series (Pete the Cat, etc). Golden Books and Berenstain Bears count, too. Basically - something that a kid will just grab and will be appealing, made very inexpensively so they can be priced very inexpensively. These are often (but not always) just crafted by the editors themselves in-house, cobbled together from existing material, bought from a packager or something, or perhaps are Work For Hire.

If you look on the Dollar Tree website, you'll see a mixture of these. Some are clearly "Real Books" but you'll notice that it says "assortment and titles may vary" -- because they don't really have control over what they get if they are buying overstock books by the pound. You'll also see mass market book-products from various places like Peachtree Playthings, Bendon, Phaidal, Dreamtivity, all of which specifically do this kind of thing, and may not normally be seen in regular bookstores.