My guess would be it's predominantly due to cryo-pumping, cryogenic propellants in the interstage liquefying and solidifying atmospheric gases, among which oxygen, because of liquid hydrogen propellant temperatures being low enough to liquefy atmospheric air on contact when Saturn V was still on the launchpad (but fuelled), so the whole interstage ring held onto it and provided oxidizer for the excess hydrogen (low oxidizer to fuel mixture) in the exhaust of the upper stage. Notice chunks of ice falling off the interstage (and the upper stage) on separation.
Let's first see what we're talking of here. From Saturn V Second Stage Fact Sheet (PDF):
The interstage, fabricated at NAA's Tulsa plant, is a semimonocoque
structure. Semimonocoque means that the skin has a minimum of internal
framework. The interstage is slightly over 18 feet in height and 33
feet in diameter. The structure has internal circumferential
supporting frames and external hat sections positioned vertically to
provide structural rigidity.
After first stage burnout and initial separation, eight rocket motors
attached equidistantly around the interstage are fired for
approximately 4 seconds. These motors, called ullage motors (an old
brewer's term referring to the gaseous zone in a tank above the
liquid), provide positive acceleration and therefore pressure to force
the stage's propellants into the feed lines to the J-2 engines. This
is called the ullage maneuver. The interstage is separated from the
second stage approximately 30 seconds after it separates from the
first stage. The two-step separation of the interstage is called
This means that the burn you see after the interstage separation can't be from the ullage motors, because they're fired to provide acceleration for the second stage propellants before the interstage separation and the video you link to shows the second stage engine has already been ignited. Plus, the ullage motors fire externally to the interstage, and the video clearly shows burning also within the interstage ring, perhaps more so than the burn external to it. The second stage interstage also separated the Saturn V first stage and the second stage LOX tanks, so the interstage provides a leak path for the LOX cryogenic oxidizer, and some of it would slush on contact with the interstage and form chunks of ice, mixing leaked cryogenic oxidizer, atmospheric air and nitrogen gas used for purging the excess gases in the interstage. Due to the nitrogen purging, there wouldn't be much of it, and indeed there isn't as seen in the video, but some there clearly is.
In a nutshell, the interstage brought along with it ice chunks of Earth's atmosphere that facilitated the burn with its oxygen where you wouldn't really expect it, once they started to outgas due to sublimation in hard vacuum and heat from the upper stage exhaust. Part of the required oxygen might also have been provided by the titanium dioxide white paint on the interstage itself, with oxygen releasing from them at the temperatures of the upper stage exhaust, or heat insulation materials used in the inner side and soaked in cryogenic propellants (cryo-ingestion of leaked propellants mixed with nitrogen used for purging them from the interstage) catching fire.
I discuss cryo-pumping and cryo-ingestion in more detail in my answer to Why was the Shuttle's LOX tank on top of the LH2 tank, since that makes it more top-heavy?