Don't copy/paste screenshots immediately in InDesign (and perhaps the same goes for Illustrator). Save your image as a PNG (if you don't mind RGB images) or as a TIFF (if you want to have it professionally printed, so it can be converted to CMYK if required). Then use the Place command in InDesign to "import" it.
If you find the images still look 'butchered', don't worry about that; InDesign uses by default a low-resolution proxy image to speed up drawing. You will find the original image quality restored when outputting to a PDF.
The reasoning not to include images in their entirety into InDesign is the following:
- including large images inside the InDesign file makes it much larger, and more likely to suddenly be damaged. (Alas. But it is an commonly accepted matter of fact.)
- InDesign works way faster without large chunks of data needing moving around.
- The original image can still be easily edited outside of InDesign. When fully embedded, there is no option to do so.