Why might the Manganese(IV) oxide and Hydrogen peroxide react this way?
Bubbles of oxygen form in the hydrogen peroxide and collect above the solution. A glowing splint bursts into flame when placed into the upper part of the beaker.
When manganese dioxide is added to a hydrogen peroxide solution (in water) is catalyzes the breakdown of the hydrogen peroxide into water and oxygen gas. The balanced chemical equation is:
MnO2 + 2H2O2 --> MnO2 + 2H2O + O2
The MnO2 acts as a catalyst here. It is chemically unchanged during the course of the reaction. That is why it shows up on both sides of the equation. The hydrogen peroxide is broken down into water (which simply goes into the solution) and gaseous molecular oxygen which bubbles out of solution and into the air. This accounts for the bubbling observed. When a burning ember (the glowing wooden splint) is brought near the beaker it bursts into flame due to the high concentration of oxygen present above the liquid. Oxygen is required for combustion. Hydrogen peroxide on its own is relatively stable in aqueous solution. The MnO2 is required as a catalyst to make the reaction occur.
I'm not quite sure, but from my knowledge oxygen is combustable and vast amounts of it was probably produced. Therefore I would assume the glowing splint had thermal energy, enough to ignite the O2 and burst the splint into flames.