"Goes" is used to indicate that something is commonly said or believed. In this case, "goes" means that the saying "You are what you eat" is a well-known saying that is often repeated.
Yes, you can write "The old saying: 'You are what you eat'" to introduce the saying. Alternatively, you could also write "As the old saying goes, 'You are what you eat'" to indicate that the saying is well-known.
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