I can't really compare non-standard analysis with filters due to not knowing much about NSA, but filters are a natural sort of completion of the poset of subsets, the pro-completion. Every subset can be naturally regarded as a filter (a principal filter), and I think of filters in general are sort of "thick" sets -- sets with some sort of infinitesimal fringe. There's also a helpful coincidence that open filters (filters on the poset of open subsets of a topological space) can be perfectly represented as filters.

It makes sense to work with "functions" whose domains and ranges are filters. From this perspective, a function X -> Y is continuous at x if it can be restricted to be a function nbhd x -> nbhd f(x), where nbhd x denotes the principal open filter from all the open neighborhoods of x.

If you're familiar with the germ at x of a smooth function f : X -> R, this is exactly the same concept as the restriction of f to a function nbhd x -> R. It's also possible to restrict to a germ at any given subset of f, which I think is neat. For example, you can restrict a smooth function to an infinitesimal neighborhood of a smooth submanifold.

These are some of the reasons why I think that filters have some "user friendliness," once you get a feel for them.