What causes oxygen cylinders to explode when in contact with oil (oiled cloth)?
Summary
⤑ Oils are mostly made up of Fatty Acids ("FAs")
⤑ FAs oxidize and release a lot of energy
⤑ The reaction is proportional to the O2 content of the gas in contact with FAs
⤑ The reaction is also proportional to the contact surface area between FAs &amp; surrounding gas
⤑ Rags offer a high contact surface area between the liquid soaked into them and the surrounding atmosphere
⤑ Maybe that increasing the surface area between air and oil (via rags) also increases volatile FAs which yields an extreme increase in contact area between FAs &amp; O2  

Tentative Explanation
I think that oils are made up of mon-/di-/tri-glycerides (at least natural oils): One glycerine bound to 1, 2 or (most often) 3 fatty acid molecules. Fatty acid chains can be oxidised, using up O2 and releasing water + a lot of energy (that's part of our metabolism, and yields much more energy per gram than the metabolism of carbohydrates). I think that the less saturated a fatty acid chain is, the more readily it can be oxidised, which is why linseed oil, which has a very high Alpha-linolenic acid content (a polyunsaturated fatty acid), can undergo spontaneous combustion. A rag exposes oil to a high contact surface with the surrounding atmosphere. Now if in addition to all of that the atmosphere is high in O2, you have perfect conditions for a high-energy-yielding reaction to massively occur.

The local temperature, oil-to-air surface area, air pressure, and air O2 content are the factors which most affect the likeliness of spontaneous combustion. The amount of oil will mostly affect how much of the reaction can occur / how long it can occur (it's the fuel in the reaction). It has a minor impact on the reaction speed only proportional to the fact that there has to be some correlation between oil volume and contact surface area.