Is this legal for transportation in the US? (video in description)
In class G airspace provided proper separation from people and buildings you should be good to go. The airspace you are in can be determined by geographic location and altitude using the FAA's charts you can find here.


  legal for transportation.


This is a somewhat broad term. The FAA may require you to get it set up as an experimental aircraft but thats up to them. If you plan to do more than hop around empty farms and over a pond you may want to contact your local FSDO for a ruling on what you can do. You can also reach out directly to the FAA for a ruling on a specific question.

The FAA as per 


  § 1.1 General definitions.
  
  ....
  
  Aircraft means a device that is used or intended to be used for flight in the air.


does define what you show as an aircraft and would most likely see it as falling under their purview. 

Note that since he is over open water he has a life jacket on board (although he is within the required 50 miles and technically does not need one). He also has a chute presumably for emergencies.