First of all, Scott Joplin is way cool. With that being said, here's my analysis of this excerpt:
To start, my harmonic analysis: Eb
mm. 77-78 bVI
mm. 79-80 V64
mm. 81 IV7
mm. 82 V64
mm. 83 V7
mm. 84 I
Okay, so here's why Joplin rocks. You see, there's this awesome technique of foreshadowing that's used in this excerpt. The listener is set up for the mixture harmony with the use of C# and F# in the proceeding measures. (i.e. C# in mm. 74) This nicely leads to the mixture that appears in mm. 79.
A voice leading pattern also makes an appearance in mm. 79 with the contrary motion of the bass and soprano parts. The G flat in mm. 78 resolves up to the G natural in mm. 79 and the C flat in mm. 78 in the bass part resolves down to the B flat in mm. 79. This stands out really nicely coming from the bVI chord in mm. 77-78. It's a nice little twist on the harmony that is both somewhat surprising and pleasing to the ear.
In mm. 79 Joplin uses a 6/4 chord to link the mixture back in with the original key. It serves as the bridge between mm. 78 and mm. 81. He also sets up the phrasing to end up with a parallel period for mm. 69-76, which gives a good feel to the line of the piece.