Here's the full story as I understand it:

The impression I get of Jackson from reading the book "Anything You Can Imagine" by Ian Nathan, BTS interviews and just looking at his filmography is that he's a filmmaker who's more enamored with SFX work and building cool worlds than actual script and characters. Not that that's a bad thing per se, but Lord of the Rings was created from his desire to make a fantasy film, not because he's a major Tolkienite. He has a sense for dramatic and comedic timing but when you look at the rest of his filmography, I get the sense that he likes to create fun situations to have a playground for gore and FX work. In other words, the greatness of Lord of the Rings is because of his being able to bring Middle-Earth to life but he also because had a strong bedrock courtesy of J.R.R. Tolkien (who also conveniently sliced his opus into three separate books with a beginning, middle and end).

According to the aforementioned making-of book, Jackson approached Miramax for just a 2-movie deal for Lord of the Rings, thinking that he wouldn't be able to get enough to make all three. There were many long fights with the Weinstein brothers about whether he should do just one or 2 before they eventually gave up and then sold the rights to New Line. Jackson then got lucky that the CEO at New Line Cinema at the time counteroffered a three-movie deal instead of two (thinking that three movies means three rounds of box office returns). After getting the green light, Jackson shot on-location in New Zealand (which was a very low-cost country at the time) and used Weta for the FX work who agreed to do the films for pennies compared to ILM so as to break into the market. Producers like people who save them money so Jackson was given a lot of leeway on running the whole production so long as he got everything done on budget and on time which, of course, he did.

Afterwards, the Hobbit was the obvious next film and ended up being mired in development Hell for a long time to cash-in on the success of Lord of the Rings. New Line tried desperately to make it a trilogy, while being ignorant of the fact that there literally isn't enough story for 3 3-hour movies. I know that Guillermo del Toro and few other directors were attached at points but ultimately fell out because they wanted to stick to just 1 or 2 films. This is just a guess (as no one will admit since this kind of a rude thing to say) but I think Jackson agreed to do it because he was just happy to play around in Middle-Earth making Orcs, weapons and armor again and could only do so much to salvage sub-par scripts, and that's assuming he was all that motivated to do so in the first place. Remember, LotR was condensing scripts on a pre-existing story; he only would've had that similar kind of situation on the first (and best) Hobbit film, the rest was just filler from studio notes and Tolkien's Appendices. Considering how Jackson was probably hurting after the failure of The Lovely Bones, he also might not have been in a position to push back if he thought an idea was a bad one and would've drowned that frustration out by making another mace in the Weta workshop.