These two chords are Bb (bVII) and E (V/vi) in C major, so on the surface that doesn't make much sense, but let's take a detour:
The Neapolitan is the major chord at the bII position in a key, and it's a predominant chord that resolves to the dominant (V). This N - V resolution is very common in minor key classic music. E.g in E minor you might hear F/A - B7 - Em. The verse of this Schoolhouse Rock tune is i - N - V - i.
The relative minor of C is A minor, and it turns out, Bb and E is a fine N - V resolution in A minor! Now in both A minor and C major the E would typically resolve to Am, but V to bVI (E to F) is a very common deceptive cadence.
In general, squeezing in progressions from the relative minor (or other nearby keys) is a great way to spice up major key music.