What does soffritto do to minestrone?
A soffritto is the Italian cousin of the French mirepoix. Both consist of small cubes of root vegetables and onions. 

The gentle “sweating” in fat enhances the sweetness of the vegetables and brings out the “umami”, an almost meaty flavor. In the onions it also breaks down the sharp pungency. The process will form a flavor base that brings a certain “heartiness” to stews and sauces.

You can skip the step - many soups will use the raw, coarser chopped vegetables - but the results would not have the properties listed above.