Can someone explain what is being said here in color publishing in the American Mathematical Monthly?

By default, color figures will appear in color online and black and white in print. If you want them to be in color in print, you need to pay the rates they indicate. All their papers appear both in print and online, it is not that you choose one or the other. They also advice on the required quality (resolution) of the figures (regardless of color) and ask that if you use the default option the colir figures are understandable and meaningful when rendered in black and white.

Other journals have an option for slightly different figures for color and black and white, which may make things easier sometimes (for instance, in black and white you use different kinds of shading as different "colors"). It doesn't hurt to ask if this is also an option here.

Color in an online publication costs nothing for them to reproduce so it is free. In a print journal, it requires a different than normal process, so the author (or preferably her/his grant) will need to bear the cost. The color pages are printed separately from the rest of the article and then assembled.

If you provide color illustrations then they can be "flattened" to greyscale and printed in the normal way so there doesn't need to be a charge for that, but not all color figures flatten properly. When they say "black and white" they probably mean greyscale, but maybe not.

The article will probably appear both in print and online. If you provide color figures, but don't want to pay the fees, then the color will be preserved online, but flattened for print.