The cross marks are used to track the camera’s movement using motion tracking software. They are just tape, but motion tracking software uses the cross marks to detect where the camera is moving. The software generates a camera file that can be imported into 3D programs and After Effects. This camera moves exactly as your real-world camera did, so any graphics or other elements you position in 3D space will appear to be in the video. It’s much more complicated than that, and takes a lot of experience to do right. After Effects has a built in motion tracker, but it’s only 2D so you can’t really do as much with it.
As for how to get rid of them, you need to mask them out. Any that the actor walks in front of will require you to hand-animate the masks.