You answered a question. Your score on your answer is greater than 1. (So this means you have 1 upvote and 0 downvotes or better. Or you could have 21 upvotes and 20 downvotes.) Just need to see the number 1 or higher for your score.
You need to have edited the question that you answered either within 12 hours before you posted your answer or edited it within 12 hours after you posted your answer.
For example, I just left you this answer, now I will go edit your question to improve it. (note: you should only make edits when you can actually improve something on the question)
OK, I edited your question. So 12 hours from now, If my answer has a score of 1 or better, I will get 1 credit for the Explainer/Refiner/Illuminator badge.
Here is a query by Jon / Martijn which will show you the count you have for these three badges. Just enter your userId (yours is 1997103 on Stack Overflow) in the input and click run.
Here is a modified query of the above one, which will give you a link to those questions of which you answered and edited to earn credit towards those badges.