Why do we want to look at the sigma-algebra generated by a collection of sets? Well, sometimes you want to know that something is true for, say, all Borel sets (see later). However, you can't really check all of those sets individually, because the usual approach of "Let E be a Borel set ..." doesn't actually tell you very much about the set E. It doesn't generally give you anything concrete to work with in order to check whether the property you are interested in holds or not. However, we do know a lot about (for example) open intervals. We may well be able to check that the property you care about is true for relatively easy sets like open intervals. Sometimes we can then boost this up to deduce that the property holds for everything in the sigma-algebra generated by our "easy" sets. And the open intervals (for example) generate the sigma-algebra of all Borel subsets of the real line. This is a very powerful abstract approach, often allowing you to prove that "All Borel sets have the following property" even though we don't have a useful concrete description of what Borel sets actually look like.
What is the (non-technical) reason for doing that? Thank you for your offer to help.
Why do we want to look at the sigma-algebra generated by a collection of sets? Well, sometimes you want to know that something is true for, say, all Borel sets (see later). However, you can't really check all of those sets individually, because the usual approach of "Let E be a Borel set ..." doesn't actually tell you very much about the set E. It doesn't generally give you anything concrete to work with in order to check whether the property you are interested in holds or not. However, we do know a lot about (for example) open intervals. We may well be able to check that the property you care about is true for relatively easy sets like open intervals. Sometimes we can then boost this up to deduce that the property holds for everything in the sigma-algebra generated by our "easy" sets. And the open intervals (for example) generate the sigma-algebra of all Borel subsets of the real line. This is a very powerful abstract approach, often allowing you to prove that "All Borel sets have the following property" even though we don't have a useful concrete description of what Borel sets actually look like.