As someone who has specifically been told, “You explain things well,” here is my number one piece of advice: Drop the pronouns and other inexact descriptors.
My mother was the queen of sentences like, “Your father and brother went to get a Christmas tree and a wreath. He left a message to say he cut his finger when he picked it up. Call him back and tell him to get one with softer bristles.”
WTF is she saying? Who cut his finger, my brother or dad? *Which * object needs to be replaced, the tree or the wreath? And she would get flustered when I asked for a clarification! A certain class of bad explainers think people can read their minds.
Here’s my number two piece of advice: Set the scene, then start at the beginning.
If certain factors are going to play into the ongoing or later action of the story, tell them up front. Say something like, “Before I tell you about my grandma’s hilarious trip to the shopping mall, I should mention she’s as deaf as a scarecrow, but she thinks her hearing is perfect.” This sets the scene for grandma’s upcoming funny exchange with the clerk.
Then give your explanation in a very linear way, starting at square one. Don’t jump around. Don’t go to the end and backpedal.
Here’s my number three piece of advice: Anticipate pitfalls in advance, and compensate for them.
For example, if you’re going to tell a story about two different redheaded oboe players named Wally, find some other distinguishing characteristics to tell them apart in your story. If the similarities have nothing to do with the story, don’t even mention them. Say, “There are two kids in my band, a tall one and a short one. The short one put a firecracker in the teacher’s locker…”
If the similarities do have something to do with the story, mention it but use something else to distinguish them. Say, “There are two kids in my band. They look exactly alike and they have the same first name. The teacher mixes them up all the time, sometimes with hilararious results. Let me tell you what happened today, for example. One of the kids lives uptown, and the other lives downtown. The uptown kid put a firecracker in the teacher’s locker…”
When you know a confusing part is approaching, go slow and carefully.
Also, anticipate your listener’s incorrect presumptions. For example, say, “Now you’d think that a man who drank 10 beers would not be able to play flawless chess, but he seemed cold sober during the whole game and won handily.”