Ordinary air is made up of only approximately $21~\%~\ce{O2}$, the primary component being nitrogen. According to the partial pressure rule, at standard laboratory temperature and pressure, this is a partial pressure $p(\ce{O2}) = 21~\mathrm{kPa}$. In a test tube filled with pure oxygen, e.g. from water electrolysis, the partial pressure of oxygen is $p(\ce{O2}) = 100~\mathrm{kPa}$ — five times as much.
According to the rate law, the combustion of wood ($\ce{C_{$n$}H_{$m$}O_{$l$}}$) with oxygen will be dependent of oxygen’s partial pressure somewhat. Thus, we can assume at least a five-fold increase in reaction rate. A five-fold rate increase also means a five-fold increase of enthalpy per time-frame which means five times the heat is liberated which means that the glowing splinter will get hotter much more quickly and finally ignite.
Technically of course, glowing is just a weaker form of burning and oftentimes only requires more oxygen to reignite. Hence why sometimes a fire can be reignited by blowing onto the embers.