The glowing splint test is used to test combustion.
In order for combustion to take place, oxygen must be present.
The flame that occurs is due to the inability to mix well with oxygen, resulting in the production of elemental carbon, glowing in a reddish yellow color.
It's important not to mistaken oxygen with hydrogen, as the presence of hydrogen will result in a "pop" sound.
A glowing splint will re-ignite in the presence of oxygen.
A glowing splint is actually used to test for oxygen.
The basic idea is that oxygen, which is a very good oxidizing agent, will support the combustion of the splint and cause it re-ignite.
The more concentrated the sample of oxygen, the more intense the flare.
The combustion reaction that takes place can be summerized like this
Check out this very cool video of the glowing spling oxygen test: