Joshua's comment to another answer, "the red copy can't be scanned" is close, but actually it is the reverse! Historically, many IRS forms used red for the pre-printed information, with the user (employer, bank, taxpayer, etc.) typing, printing or hand-writing the actual information. This makes scanning by the IRS easier. (It has nothing to do with scanning forms into your own computer.)
The problem with printing the red page of the downloaded PDF is that most people don't have a color printer capable of doing so properly. Even a good color laser printer may not print the shaded areas properly, and the color - which is detected by the high-speed scanner - may end up significantly different than intended. Printers (that's "businesses that produce printed material", not "little boxes made by HP") will actually use a red ink, typically formulated to a particular Pantone (or other very specific/defined color) color to produce consistent results. With a typical ink-jet printer, the results are even worse, particularly if one of the ink cartridges is a little low when you print, or if there is any smudging due to wet ink or glossy paper. With a typical desktop printer (color or laser), all colors except pure cyan, magenta, yellow or black are produced by a combination of cyan, magenta, yellow and black. This is a 4-color printing method. But real printing presses can also use specific pre-mixed colored ink. Typical IRS forms would be printed with only black and the specified red - 2 colors instead of 4.
I would not be at all surprised if the PDF, printed on a high-quality color laser printer would work 100% fine - scan by the IRS with no errors. But I wouldn't count on it, and the IRS is telling you don't even try.
Thanks to dave_thompson_085 for the IRS publication that gives the specific ink: Flint J-6983 red OCR dropout ink or an exact match. The same ink is commonly specified for health insurance claim forms, which have a similar history of high-speed scanning long before everyone had a $100 printer/scanner on their desk.
I found one reference to the red being an anti-copying measure. While that could be true, I don't think it is the primary reason. After all, you could receive a completed 1099 (or other form) and white out and overtype the contents and it would not affect the scanning significantly. Plus most of these forms are readily available, so you could make up your own from scratch. The biggest anti-forgery (i.e., copying to somehow mess with the information) feature is likely (or at least should be) that employers send information separately (electronically in most cases), which can be matched against the information sent in by individual taxpayers. Any discrepancy should flag the return for verification.