OR is used to broaden a search. If you connect two or more terms with OR, either of those terms can be present. They might both be present, but they don't have to be present like they do with AND.
Let's return to the burger database. Say you're just not finding that many burgers with ketchup on them. Something must be wrong. Aha! Ketchup is sometimes spelled "catsup." You can use OR to find both spellings.
Once again, a venn diagram may help to explain:
In this case, the search would include the middle, overlapping portion, where "ketchup" AND "catsup" are present, and the entire "ketchup" circle and the entire "catsup" circle. So, you may have some burgers with only "ketchup" on them, some with only "catsup" on them, and some with both ketchup and catsup. I, myself, am fond of catsup.
In other terms, OR is useful when you can think of synonyms, alternate spellings, or other ways of phrasing a search. A few other examples:
cannabis OR marijuana
bipolar disorder OR manic-depression
dogs OR canines
humor OR humour