What is Katakuri's power?
His power comes from a devil fruit, a "special" type of the paramecia called Mochi Mochi no Mi. Although this fruit is a paramecia type, so far its characteristics have been more inline with the Logia, as the user can become "intangible" to avoid attacks. An example is when Ichiji tried to hit Katakauri
 

Also worth of nothing is that in the first translations, the fruit type was labeled as Logia but was later changed to a special type of paramecia(i would say because mochi is not a natural element).

Regarding to his ability to predict the future, its just a further developed observation haki. Usually this type of haki was shown to enable the user to perceive what kind of attack the enemy will use, so its a natural extension to be able to predict the movement of your surrounding in the future.

Info about mochi mochi no mi:
http://onepiece.wikia.com/wiki/Mochi_Mochi_no_Mi

Wiki about mochi, if you want to eat some cake 
https://en.wikipedia.org/wiki/Mochi