Basically, the positive anymore does not simply have the meaning of nowadays, but rather means simply quite the opposite of negative anymore. The negative anymore implies that what is described by the sentence used to be the case, and asserts that it no longer is, the positive anymore implies or asserts that what is described used to NOT be the case, and asserts that it is now.
Kindle and Sag (1975) provide a slightly more technical explanation. Consider the following:
(1) Anymore, we eat a lot of fish.
According to Kindle and Sag (1975):
The usual hypothesis advanced about
the grammars of those, primarily
Mid-west, speakers who say sentences
like [(1)] is that they have
restructured anymore into a
free-wheeling lexical item with the
meaning of 'nowadays'. [...] This
explanation has recently been shown to
be unsatisfactory by Labov (1972), who
observes that all English speakers
balk at items like [(3)] and [(4)].
(Kindle and Sag 1975:89)
(3) When would like to live, 1920 or
(4) When was the best beer brewed? ...
Kindle and Sag continue, quoting Labov (1972):
'In Standard English a sentence of
the form: 'I don't do Y anymore'
presupposes that 'X used to do Y'. In
these 'positive' anymore dialects a
complex semantic change has taken
place creating a new lexical item
anymore-2, which occurs only in positive sentences. Positive sentences
of the form: 'X does Y anymore' assert
that 'X didn't used to do Y.' Positive
anymore speakers still have the old anymore in negative sentences, i.e. as a polarity alternant of still.'
(Labov 1972, cited by Kindle and Sag 1975:89-90)
Kindle, D. and I. Sag. (1975). Some more on anymore. In R. W. Fasold and R. W. Shuy (eds.), Analyzing variation in language: Papers from the Second Colloquium on New Ways of Analyzing Variation. Washington, D.C. Georgetown University Press, 89-111.
Labov, W. (1972). Where do grammars stop? In R. Shuy (ed.), *Georgetown University Round Table on Languages and Linguistics 1972. Washington, D.C. Georgetown University Press.