The answer to this question is relatively simple. Students use academic writing services because such services are extremely helpful to them in their studies. How are they helpful? The answer to that question is a bit more complex.
First and foremost, model essays are an excellent way to learn how to write. I've said before, on many occasions and to many people, that it doesn't make sense that we give students models in math (in the form of sample problems) and models in science (in the form of the instructor showing the students how to do the lab before they do it themselves), and yet never in writing. In writing, particularly when students pass the high school level, they are simply expected to know how to write, and write well.
This does students a disservice, because the reality is that many of them do not write well. This is not an insult. It's a fact. I can write well, but I can't landscape a yard, or fix my brakes. That's also not an insult, but a fact. So, for those students who struggle with writing (and who always may struggle with it), a model essay on a particular topic, written by a professional writer who knows what s/he is doing, can really be a guide to them in writing their own paper.
Second, model essays can help teach students how to conduct research. By reading a model reference page, for example, students will see that Wikipedia doesn't appear on the list (or, rather, it should not). Students also get a good sense of what should be on that list (e.g. scholarly journals and books). The model essay will also show how the research is used to write the paper – how quotes are used, how material is paraphrased properly (meaning, with correct citations), and how to blend in researched material with original thought. This sort of integration of what we learn from other places and what we learn from thinking about things is a very tricky endeavor, and so having a model from which to learn how to do this can be very useful.
Finally, academic writing services save students precious time. Today, most students are not the traditional, classic picture of student that we have in our collective minds when we think of college and university students. Instead, students are married with families; they are holding down sometimes two or three jobs; they have ailing parents for whom they are the primary caretakers, or children with autism, or any one of a large number of other heavy responsibilities. Far from blowing their time on frat parties, today's students are more likely to be burdened with work and family demands upon their time. So – is there any reason they should not lighten their load? After all, they aren't cheating by using an academic writing service. They are purchasing a model, a guide, something that will help their own writing process go much faster. Can we blame them for doing that?
There are other ways in which academic writing services are helpful to students, of course, but these are the top three. And being helpful to students is why they patronize model essay writing services. Such services – those that are legitimate, that provide original material, and that do not rip off their own clients – are not cheap. They come at a price that is commensurate with other valuable services and products. Therefore, students, especially those on a budget (and who isn't?) would not use such services if they were not worth it. These are some of the reasons why.