Here's what happens here.
Keep in mind that carbon dioxide itself does not change the color of dry litmus paper.
However, a solution of carbon dioxide will turn blue litmus paper red, which implies that a carbon dioxide solution is acidic.
You can get the same result by exposing carbon dioxide to moist litmus paper--the gas will react with the moisture to produce an acidic solution, thus turning blue litmus paper red.
The idea here is that when you dissolve carbon dioxide in water
#"CO"_ (2(g)) rightleftharpoons "CO"_ (2(aq))#
you get an acidic solution because some of the dissolved carbon dioxide will react with water to form carbonic acid,
#"CO"_ (2(aq)) + "H"_ 2"O"_ ((l)) rightleftharpoons "H"_ 2"CO"_ (3(aq))#
So depending on the actual concentration of carbon dioxide in solution, the