A couple of remarks:
In Zentralblatt MATH "keywords" are explicitly included. For example, see this recent entry for one of the OP's papers (link should work without subscription; the precise choice of the paper was random). And one can search for them (under category 'basic'); possibly 'browsing by' is the more interesting way to use this as discussed in the this question on searching related to MSC classificationquestion mentioned above. [For similar remarks for MathSciNet see Patricia Hersh's comment.]
This is more for books than for journal articles, but (certain) electronic library catalogues contain "keywords". These then can also be used for searching. I occassioally found this useful. Say, I search for a book on some subject; I know one, I go to its entry, I click on a keyword, and get a list of books with the same 'keyword'. These keywords than can also be in some hierarchy and inter-linked. Personally, I never had a need to use this extensively as I was (essentially) always close to a math library that porvides free access to the books (and had the books ordered by rough subject), so that browsing the hard-copies seemed more efficient; however if I ever should be in a situation that the library does not provide free access but one can only order books, then I think I could find this option quite convenient. Say to figure out which (text)books on a given subject are available in the library. (True, often the title can serve as a proxy for this, but it does not always.) Furthermore, I think that in cases of publications in different languages 'keywords' are useful to have just some simple and short bits of information (the keywords) to translate that then already allow to link a publication into a body of literature. [This is not done by us (mathematicians) but those that need to do it depend on us to provide meaningful information to process, so that then later we can find the things]
The point that sometimes well-chosen keywords can convey something on the content of an article that title and abstract do not was already mentioned. I agree.
I think, but do not know for a fact, that meta data of articles (like keywords) is useful/used in order to make general search facilities work better. So, it might be that even if one does not search via keywords directly the fact that some general search mechanism (say general purpose search engine) will work well and turn up the article(s) one is interest will among others depend on the fact that somebody provided some good keywords and they were than incorporated in the electronic resource in a suitable way. Also, some electronic article repositories do suggest related ariticles; I think here also keywords might be used to figure out what is (supposedly) related. [Personally, I so far never found this feature very useful, but then I have to admit it never really paid much attention to these suggestions even. Perhaps I should.]
So, in brief, I sometimes found keywords useful (though not very or in an essential way). Yet, I think to a considerable extent this is due to the existence of the MSC and if it would not exist they would be considerably more important [yet as was pointed out already that MSC works well depends to some extent on keywords]. Also, MSC is 'only' for math so in cases where publications of differing subjects all need to be dealt with (like a general library catalogue, on the web,...) keywords become more crucial. And, some of their use might actually happen 'behind the scenes'. Yet, likely, it is true that to some extent keywords become less relevant as fulltext search and so on becomes more feasible (but this is a reletively recent phenomenon).
Thus, for the question how to use keyword, I would answer for example for searching/browsing where this is supported like Zentralblatt MATH. And, as already mentioned, to get some quick idea what is inside the artcile.
And, for the possibly present implict question whether one should care as an author to provide meaningful keywords when asked to, I would say definitely yes; it might be indirectly of use in ways one does not fully oversee and is not that much work. (And if one does not provided them, somebody whose job it is to add them somewhere might just pick something that seems alright, with best intentions for lack of a different option, but might chose something quite wrong.)
Finally, since we are on MO, the super-brief version:
Keywords are just tags by another name.