How is he playing a D7 in the key of C major?

It's such a common occurrence, it's taken for granted. Here, in key C, D7 is V/V - the dominant of the dominant. The D7 leads diectly to G, which then does the same into C.

1. Is D7 in the key of C?
2. What is v7 of C major?
3. What key does D7 belong to?

Is D7 in the key of C?

In the key of C Major, D7 is a secondary dominant chord; a brief rationale for secondary dominant chords could be as follows: they make use of some note outside the key (in C Major, either # or b) to bring a change in the function of the chord.

What is v7 of C major?

The C major chord V⁷ is the G dom 7 chord, and contains the notes G, B, D, and F. This dominant 7th chords root / starting note is the 5th note (or scale degree) of the C major scale. The roman numeral for number 5 is 'V', and is used to indicate this is the 5th chord in the scale.

What key does D7 belong to?

Chart with four-note chords sorted by key