Let us define a student to be someone who makes a study of something. Furthermore, allow study to be defined as the active pursuit of knowledge. For instance, someone who studies because they crave the acquisition of a particular knowledge is doing so actively, whereas someone who studies because they are required (i. e. the college student who is attending college solely for the purpose of a degree) is not. Integrating the two definitions, we deduce that a student is someone who actively pursues knowledge of something.
Let us also assume that the role of a teacher in any student-teacher relationship is to impart to the student the knowledge he or she desires. Consequently, it is necessary that if the teacher is to teach the student, the teacher must possess a superior knowledge of the subject the student is pursuing. Although a superior knowledge is necessary, it is not sufficient. A teacher must also be able to aid or facilitate the student in his (her) pursuit of that knowledge. This, by definition, makes the teacher who enables the student to acquire that knowledge in the most efficient manner the best. Hence, the statement, students are their own best teachers is tantamount to saying that the most efficient way for a student to obtain a particular knowledge is to teach it to themselves.
For instance, suppose students A, B, and C are taking a college level general biology course with their intent being the acquisition of knowledge. Although there is only one individual who is generally referred to as the teacher (the person giving the lecture), he / she is not the only thing from which these students can acquire knowledge. Certainly, the professor and the textbook possess more biological knowledge than the students do. Moreover, often times there are other students who have previously studied a form of biology and possess an amount of knowledge that these students lack.
The point is that any of th three sources are capable of teaching the students. However, from which source the student acquires knowledge in the most efficient manner is dependent upon the student. Suppose student A learns best (most efficiently) from lecture, student B from the textbook, and student C from his roommate. Suppose also that student A does not attend lecture, student B does not read the text, and student C does not converse with his classmates. Although each student has the potential to acquire this knowledge in the most efficient manner, not one makes use of it. Therefore, it is it is left to the students to decide which teacher (professor, text, or classmate) will best enable them to learn (acquire knowledge).
So regardless of the teacher, it is, and always will be the students who enable themselves to acquire the desired knowledge in the most efficient manner. This, by definition, makes students their own.