I agree with nozomiiin's explanation, but not with mfuji's.
解説 has a narrower meaning than 説明. 解説 refers to a logical and detailed explanation/lecture/commentary about difficult concepts, problems, etc., usually given by someone who is knowledgeable about that topic.
- 教授に相対性理論についての解説をしていただいた。(相対性理論 = theory of relativity)
You can usually replace 解説 to 説明, but 解説 is often preferred because it sounds more detailed and authoritative.
In sports, a main play-by-play commentator is called a 実況 (who is usually a professional announcer), and a former athlete who sits next to the 実況 and provides expert analysis is called a 解説(者). This is a fixed role name that cannot be replaced by 説明(者).
説明 is explanation in general, and it doesn't have to be related to expert knowledge. You can only use 説明 in the following situations:
I don't think 解説 is more subjective and 説明 is more objective. Contrarily, 説明 can be more personal and subjective, whereas 解説 is at least supposed to be unopinionated and fair.
解釈 is "interpretation" of something difficult or ambiguous. Basically it's an internal process done only in one's head.