One trouble is that the "idea" you've just communicated isn't actually the idea behind linear regression. The idea behind regression is this.
You're trying to make a prediction for some number you care about - let's say the value of a given stock price. You've come up with a set of hypotheses about which characteristics might help you make that prediction. Moreover, you have a set of examples that you've witnessed in the past, and you want to learn from that experience.
Using linear regression, you can test those hypotheses. You turn those characteristic features into a quantifiable number themselves. Linear regression is simply the name we give to the process of testing whether there's any validity to your hypothesis. If that hypothesis is true and you've discovered what makes the stock go up, then the corresponding feature will be given a high absolute coefficient. You'll also know whether it's an indicator of the stock going up or down, based on the sign of that coefficient. There's no math involved - you're testing your own intuition about how to make predictions.
The idea behind linear regression isn't "finding a line that fits a scatter plot." That's still math and it's still unhelpful to many people with serious, real-world applications. It's just an abstraction of the math that happens to leave out the formal representation.
To really communicate ideas in application, you need to move past the math entirely, and get to how it ties into people's judgments about data that they know well and have experience with, and show them that the intuition they can bring to the table is valuable (for feature determination). Otherwise, even scatter plots will often shut people out, because they "aren't good at math."