As other answers state, the effect is called Moire. But why does it happen when you downscale or zoom-out? As prevoiusly stated Moire happens when two patterns interact, specially if the two patterns have a "frequency" (read size of the repeating characteristic) close enough to each other.
What happens next is a mathematic relationship between the patterns, or more precisely the sizes of the patterns: Most likely your phone or camera has higher resolution than the screen you are caturing, so the lens projects the image of the screen's grid into the sensor's grid, but each pixel from the screen is projected much larger than each of the sensor's. This means that the pattern from the screen is much larger than the sensor's grid. Think of it as laying a grid fence panel over mosquitoe screen. You would not notice moire so much in this case.
But then you downscale the image, or zoom out. This is actually reducing the size of the pattern originally captured from the screen and bringing it to a scale much closer to the second screen (the one you are watching in). It would be like shrinking the fence grid from the prevois example until it is almost the same pitch as the mosquitoe screen. Now that the two patterns are very close in size the moire is much more noticeable.
Regarding how to remove it: I think that there may be a way of getting (almost) rid of this moire. And it may be applying a certain amount of gaussian blur and then downscaling the image. The idea is that the blur will fuse toghether the image from the original screen's pixels, practically blurring the line between pixels. This will also blur the border of letters and graphs in the image, that's when the downscaling comes in. Downscaling tends to hide blurriness. Hopefully it wil produce clearly legible letters and graphs but hide the pattern from even areas of the original image. By removing the original pattern, there wont be two patterns interacting and the image should be more legible.
I want to perform an experiment for this, to add it to this answer later. Please remind me if I forget. (give three days).