How can I prove my self-studied knowledge?
One may prove their knowledge by these aspects.


They may have a certificate of the courses they have passed.
They may have a publication or patent registration in the field of their personal studies.
They may have worked in the field of their knowledge and their projects and portfolio is a proof of their knowledge.


A website designer may not have any certification of their design knowledge, but the websites they have designed are the proof of their knowledge. 


You may have done voluntary/paid teaching at your undergraduate university or in an educational institute, so you have the certification of your teaching activity and that may be counted as a proof of your teaching and your knowledge. (Also teaching/research assistantships to a course which you have never passed but you did assistantship in that area.)
You may have some publications in the area of your knowledge, for instance, a published paper or book; or a contribution to a publication which is so related to your knowledge.
You may have done some jobs related to your knowledge. An engineering design, done some code-developing, etc.
You may have a recommendation letter from a professor in which has written that you have sit in their class for that course, but you have never registered for the course; so it proves that you have the knowledge of the course.


However, as far as you are registering for a graduate program, you must fulfill their requirements not what is generally/logically reasonable or what seems to be acceptable. So it is better to provide all your proofs in your CV or their application process website and let them ask you for more official documents. Also, you can email them and ask for your special issue.

P.S. It seems that your question indicates self-study knowledge, but some people may have done some non-degree programs and certificates of those courses may also prove their knowledge in the course.