This sequence of chords also appears in Hoagy Carmichael's "Two Sleepy People" and Fats Waller's "Ain't Misbehavin'". Waller used it quite a bit, in fact. It also appears in Charles Trenet's "La Mer" and Pete Seeger's "Where Have All the Flowers Gone".

One can interpret it as just being a cadence, repeating over and over. Am7 has the same notes as C maj 6 and Am (with or without the minor seventh) can be interpreted as a "substitution" for a C major chord. F is the relative major of Dm, so one could play C Am Dm7 G (resolving to the next C in the cycle). Playing G7 emphasizes that it's the dominant. One could just play C instead of C Am or one could play C maj. 7 - Am7 Dm7 G7.

One can vary the pattern by using Em or Em7 to substitute for the G7 chord.

The trick is to notice which notes are shared among chords.

Other songs use the more simple cadence F G C or Dm7 G7 C maj. 7 (or C6 or Cm or Cm7, etc. ), a.k.a. "ii - V - I (or i)", repeatedly. Sometimes they will change key occasionally and there will be, say, Em7 A7 D maj (or D min).

The variations on the cadence are the most common idiom in Western popular music, followed by smaller or larger fragments of the circle of fourths. In fact, the cadence ii - V - I is a fragment of the circle of fourths (G is the fourth of D and C is the fourth of G).