Transport moving object/person to another video
The answer you're looking for is called "masking."

You can find this tool in the majority of professional editing software, but I personally recommend adobe after effects. There is also a chance that some of "moving objects/people" are already presented on green screens-- where as masking would usually be necessary, a "chroma key" would instead take the job.

Simply "cutting out" the moving figures isn't the only technique seen in this video. If you plan to execute a video in this fashion, you'll need to both cut out the "moving figure," but also motion track these figures onto the new composition. From there you'll need to manipulate their positions in order to make a convincing and visually appealing effect.

Good luck!