I just want a simple but comprehensive way of graphically showing standardized beta weights as predictors and a way of show how much variance our model explains
Don't do it! Standardized beta weights and variance accounted for will be meaningless to clients. I recommend showing the effect in real terms - "by implementing program A, the risk of contracting the disease is reduced by 9%" or "for every 5 g increase in sugar content, taste satisfaction increased by 0.7 points on a 9-point scale" or "eating a low carb diet for 10 weeks resulted in a loss of 6.3 pounds."
Concrete statements in real terms will be so much more meaningful for non-statisticians. If you can plot the relationship between real variables, not z-scores, the client will appreciate it. If you can supplement these graphs with measures of uncertainty, then you can educate them regarding the range of possible outcomes. It's up to you to know that what you are reporting is "real" (i.e., not due to chance).
I concur with Michael's opinion, above. I would only add it is sometimes helpful to include other variables in a multivariate regression. Building on Michael's example (above) "eating a low carb diet for 10 weeks resulted in a weight loss of 6.3 pounds, and adding a 10 minute walk resulted in a further loss of 3 additional pounds, while drinking a cup of coffee with the other two had no effect." Otherwise, I cannot add to his advice, it is excellent
You might want to show your clients a comparison, on the same graph, of y on the y-axis and predicted-y on the x-axis, for competing models. (You don't want to overfit, so you could save out some data for validation.)
Perhaps something in the following will give you an idea, though it sounds like your data may differ from the kind of data in my examples: