There is no standard method. Why not? because outliers are different things from what you have normally, and vary for each problem. They come from a different distribution than the normal noise in your data.
Here, it appears that you have a relatively large fraction of outliers, in comparison to the data. Effectively, you don't have outliers! That is just your usual noise.
You ask what is the standard method of plotting data without outliers? You find them. Then you plot only the data that was not an outlier. What else would you do?
There is an rmoutliers tool, in the stats toolbox. But if you have a LOT of outliers, and relatively few, noisy data points, then it won't be of much value.
You can use tools that are designed to work robustly. robustfit is one such tool, again in the stats toolbox. These tools are usually in the form of iteratively reweighted estimators, so they look for data points with large residual to the model you pose, and then downweight them in the next iteration.
Finally, the above comment notes that you need to choose a model. They cannot do that for you. The "curve of best fit" is a meaningless thing, UNLESS you have a model in mind.
