I propose the following way to introduce filters on sets to beginning students. (I am writing a book which contains this intro now.) You are welcomed to comment whether this is a good exposition and how to make it even better.
We sometimes want to define something resembling an infinitely small (or infinitely big) set, for example the infinitely small interval near on the real line. Of course there are no such set, just like as there are no natural number which is the difference . To overcome this shortcoming we introduce whole numbers, and becomes well defined. In the same way to consider things which are like infinitely small (or infinitely big) sets we introduce filters.
An example of a filter is the infinitely small interval near on the real line. To come to infinitely small, we consider all intervals for all . This filter consists of all intervals for all and also all subsets of containing such intervals as subsets. Informally speaking, this is the greatest filter contained in every interval for all .
[A formal definition of a filter on a set goes here.]