It shows a method or "way" of foods making a difference. In other words, if you eat bread and cheese and get fat, then you can say that the food shows you how you get fat (or the experiment, but I'll get to this in a second).
If you are to break it down by structure, it would be like this:
"It shows." This is one part of the sentence. It can be separated by taking out the "how."
Then it can be connected with a "that," becoming, "It shows that it's often the everyday foods like cheese and bread that make a big difference to the amount of saturated fat and salt we are eating."
If we are to change the original sentence but keep the meaning intact for easier understanding, you could take the "how" and put it in the back, along with the rest of the beginning.
"Often every day foods like cheese and bread make a big difference to the amount of saturated fat and salt we are eating is how the experiment shows it."
If you understand the meaning of "how," then I don't see why you would have trouble in understanding this sentence.
Last example that is unrelated to your sentence:
"He is doing magic."
"He is doing magic. We learned to do magic, too."
"Him doing this style of magic is how we learned to do magic." <--- in this case the "how" follows the "is" because it modifies the way something "is."