Is there any program that allows a user to read eBooks or PDFs in a way that does not bother the eyes, pretty much like what Kindle does? Maybe by reducing the illumination of the monitor and using some other tricks?
If you are experiencing eye fatigue reading the screen on your Mac, you should try the settings available in System Preferences: Universal Access under the "Seeing" tab. In the "Display" section, you can invert the screen from the usual "Black on White" to "White on black", giving everything a black background with white text, and inverting all the colors. You can also choose Grayscale, which will eliminate colors altogether, and you can adjust Contrast.
What you are asking is impossible, I'm afraid. The Kindle uses a different type of technology for its screen called "e-ink" or electronic paper:
whereas laptops (and iPhones, iPads) have backlit LCD screens.
The essential difference is that the Kindle screen is lit by the light reflected off it - the same as a printed page - this makes it more relaxing on your eye to read.
There is no software way of achieving this on a Mac - you'd need an entirely new screen!
There have been rumours that Apple own some e-ink patents but nothing has come of this so far.
The Mac app
f.lux will automatically make your screen dimmer and more orange, and thus easier to read at night. This also prevents blue light on the screen, which might help you sleep better. Download F.lux.
You may want to try Nocturne (last app on that page). Nocturne is no longer available, see below for a forked version.
Among other things, the program lets you remap the black and white points of the colors on the screen.
While the default options are pretty extreme, you could do something like map white to a bright grey, and black to a dark grey, resulting in less contrast. I prefer to use it with black set to a dark grey, and white set to a shade of blue or red… I find it a lot easier on the eyes.
Nocturne is no longer available, but there is a fork named Tranquility. Unfortunately, it hasn’t been updated since 2015 (as of February 2017), and there is a comment on this issue suggesting that it was broken by 10.11.6. I have not personally tried this software, so download at your own risk.