How to separate an image to layers for separate cutting

If it is a vector image, just select all the objects of one color. Then export and set it to export only the selection. Repeat for the other color. Then you have two files that you import into the cutting software.

Some cutting software will separate the colors for you, for example SignCutPro.

If it is a raster image, it could not be use for cutting to begin with. It will have to be traced, either manually or with auto trace.

Then do the above steps on the resulting vector image.

Edit:

It occurred to me that you might be using the cutter with a print driver.  I don't like solutions that involve sending data directly to the cutter via a print driver.  Dedicated cutting software often gives you more precise control.  However if that is the case, you can select all the objects of one color and move then to a new layer.  On the layer docker you can click the little print icon, that toggles whether the layer will go to the printer or not when you print.