There are a few reasons that Self-Determination theory works and that there are educators that subscribe to this psychological idea of how to motivate students.
In regards to Self-Determination Theory there are many causes of how students are motivated and choose to work harder in certain classes and harder on certain projects. The three main causes for what motivates a student are the ideas of competence, autonomy, and relatedness these three are all pieces of intrinsic motivation that allows for student to develop a sense of self-worth and self-learning (Deci, E. L., Koestner, R., & Ryan, R. M. (2001)).
The idea of competence is the least important of the three according to the research the purpose for competence is that when a student is doing work if they feel like they are competent they will be more intrinsically motivated to keep going. If the student understands the material then they will care more about it, because if a student understands a math problem then they will want to continue doing math, as opposed to a student who does not like math they will develop a mentality of being bad at math forcing them to no longer want to learn. There are ways to overcome a student who does not care about the class because they are bad at the class.
Relatedness is a very difficult task to create in a secondary classroom setting where there are 150 plus students a teacher sees everyday, but that does not diminish its importance. Relatedness comes when a teacher develops a relationship with the students (Deci, E. L., Koestner, R., & Ryan, R. M. (2001)). If students like their classmates then that helps foster a relatedness to the subject, but developing a relationship the teacher is key. Once a student feels a relation to the teacher then a relation to the subject can be created. This will allow a student to identify with the subject which makes them care and causes them to work harder in the class, causing self-determination theory to hold weight in the educational community.
Autonomy means that a student will be able to have some sense of control in the classroom or how they are going to do an assignment. Building a new sense of educational value in a student because they want to learn about what they have a say in. The new causality of their learning is important to foster and create a spark of drive in the students. Every time a student gets to choose what they research or help to decide the classroom rules it gives them a sense of autonomy a sense of freedom in the classroom which makes student work harder to achieve the goals the teacher has set forth. When these three pieces are brought together there in itself are the three causes for why Self-Determination theory works and how you psychologically motivate a student to pay attention and care about their class (Deci, E. L., Koestner, R., & Ryan, R. M. (2001)).
Educators undermine intrinsic motivation on a regular basis because they want to maintain total control of the classroom and do not want to relate to the students. Regularly teachers do not want to attempt to motivate the unmotivated students or put effort into their under-preforming students, often those students will be written off showing preferential treatment to the "smarter" students. Once competence is compromised it becomes difficult to bring a student back into the fold (Iyengar, S. S., & Lepper, M. R. (2000)). There are many forms of external motivation and some of those patterns work because there are so many different ways for student to derive motivation, but the system of rewards for actions can cause student to no longer care about the work they are doing. If the only rewards that are offered are very external such as a piece of candy then once that is removed either later in life or in a different classroom that teaches the same subject the student will no longer care about the material. So if an educator can externally convince them that there is a reason to follow up on the material then there may still be an external motivator such as being more well rounded as a student, but it will create a better lifelong learner out of the student who needs the help in order to grow and learn (Iyengar, S. S., & Lepper, M. R. (2000)).