It's a bit difficult to explain, but I'll try to answer. "解説" is included in "説明" . "説明" means to explain something to make it clear or to comment on an idea. For example 医者が（病気の）症状を説明する： When you see the doctor, the doctor explains your symptom. 仕事に遅れた理由を説明する：When you arrive late to work, you are required to explain the reason. When it comes to "解説" , it needs a kind of analysis with specialized knowledge before explain. For example (石油価格高騰についての)ニュースを解説する：To analyze and explain the reason and background of the news for surging of oil price.　To comment on the news for surging of oil price. アイススケート（競技）の解説をする：To explain the technique and comment on the wonderful play of skaters in a figure skate competition. Hope this will help :).

解説 is a deeper take apart (including 説明) to deeper issues a 説明 is mostly simply a quick guideline mr.smile has a good demostration by "symptom of illness" vs "the news for surging of oil price" but for something more comparable, you may want... "The how-to-play of tic-tac-toe" vs "The how-to-win of tic-tac-toe" because quite often, these two terms are used alone like 説明：to have a line of three of yours to win 解説：to win, you must not place your piece beside the center-first-move of your opponent. the reason is that, it will make your ....(etc for a few hundred words)