As a naturalized American, I usually say, "the United States" or "the US", but sometimes I do say "the States" as well, given that this seems to be a relatively popular choice... at least on the Internet. How did the expression "the States" rise in popularity?
While the OED's earliest examples of States could be argued as referring to the individual states, in plural, it has an entry from 1846 from Richard Bonnycastle's Canada & Canadians:
Irish canal men.. cross over from the States to Canada
which seems pretty clearly a reference to the country as a whole. The earliest attestation from a U.S. source is 1945, from the Richmond Times-Dispatch:
The Forty-fifth Division's ‘most shot up soldier to return alive’ is back in the States.
In either case, the States is likely to have been in informal use for some time before its appearance in writing.
For what little it's worth, I have heard the Americans and foreigners alike refer to "the States" my entire life (which begins in the mid-1970s), and anecdotally speaking, I wouldn't say it is any more common now than before. If I may posit a theory, however, the Internet (not to mention 9/11 and subsequent adventures) have made it more difficult for the average American to tune out the outside world. It may be that what you are hearing is more references to the country in general — as the U.S., America (and 'Murica), the United States, as well as the States. Even in my circles in Washington, however, which are by most standards very highly educated and very well-traveled relative to most of the country, Americans talking to other Americans just say "here" or "at home," like the trains in Japan are more reliable than the trains here and almost never the trains in Japan are more reliable than the trains in the U.S. or any such.
The British North American colonies occasionally were called states as far back as 1630s; the States has been short for "the United States of America" since 1777
And from the OED:
b. In plural with the and capital initial. Originally: the states (sense 29a) of the United States of America considered collectively. Later (colloquial): = United States n. 2a.
1776 R. H. Lee Let. 21 July (1911) I. 210 As well for the honor of Congress, as for that of the States.