The term 'Degree' is used to identify a particular note in a scale. The point of using this terminology is that it can make life a whole lot easier when trying to identify chords and intervals in new keys.
As outlined in the video. If you take any scale and begin on the first note of that scale, (in this case it is the C major scale) that note will be the first degree. The second note is the second degree, the third note is the third degree, and so on.
Once we reach the 8th degree, the scale begins again. We can continue naming the notes from here as scale degrees up to the 13th degree. At which point we stop.
The reason for this will become clear as we progress and begin building chords.
Have a look at the diagram below for an example of how we can name scale tones as 'degrees'.