Why is liquid oxygen more reactive than gaseous oxygen?
Since oxidation is generally strongly exothermic, a large part of the difference in reactivity can be explained by kinetics (rather than thermodynamics): the denser liquid brings oxidant (oxygen) into intimate contact with the oxidized substance throughout the reaction. Consumed oxygen is quickly replenished by surrounding liquid oxygen. In the gas the exothermic nature of combustion reactions accelerates reaction with oxygen gas by increasing the collision rate between oxygen molecules and the surface being oxidized. In the liquid this temperature effect is not as important because the oxidized surface is already bathed in oxygen molecules. Also, although the gas might be at a higher T, as assumed in the question, local heating of liquid oxygen around the oxidized surface due to the highly exothermic nature of most oxidation reactions can also add to the faster propagation rate of reactions in the liquid. The importance of thermal effects might also be reduced (if not trumped) by the acceleration of concerted events during the oxidation (such as sequential events at the same or neighboring sites or while in a short-lived activated state), since in the liquid collisions between surface and more than one oxygen molecule at neighboring sites will occur at rates that are orders of magnitude faster than in the gas. 

This is admitedly an answer in need of some quantitative support.

Here's a "back-of-the-envelope" justification for the importance of density:

It turns out that if you use a model based on the Maxwell-Boltzmann kinetic theory then the collision frequency with a surface is proportional to $p/T^{1/2}$. Therefore, if you ignore the thermal activation energy for reaction, you can assume that the reaction rate is proportional to $p/T^{1/2} = \rho _m RT^{1/2} = RT^{1/2}/V_m$. This means we will have the same collision frequency for two different set of values of T/V when

$$\left(\frac{V_{m2}}{V_{m1}}\right)^2=\left(\frac{T_{2}}{T_{1}}\right)$$

Therefore, we can counter the effect of a 200 K drop in T by increasing the density (reducing the molar volume) by a factor of ~14. Now at STP the molar volume of a gas is ~20 L/mol, while for a compressed (liquified) gas the molar volume is 100 times smaller, say 0.2 L/mol. Therefore the increase in density more than makes up for the drop in T, at least within this simplified model. 

In more general terms, we can explain away much of the loss of the kinetic advantage of higher temperature by the higher density in the liquid. So based on this simplified argument oxygen at a liquid-like density can be more reactive than a much warmer gas.

Note that the activation energy for reaction of $\ce{O2}$ is not negligible due to the relatively stable structure of the oxygen molecule. Therefore kinetic effects involving rapid heating and thermal activation of oxygen (due for instance to local accumulation of thermal energy) are probably also important to explain the reactivity of LOX.