Fuel + air + ignition = firef and/or explosion.

If you have fuel (oil) and air (oxygen) always present, then if an ignition source comes along, you don't want to be there.  Hence, by keeping air and fuel separate, two events have to happen at the same time, the fuel and air have to be there AND an ignition source has to be there all at the same time.  The probability is simply reduced.

There are some other factors such as the right amount of fuel (between the upper and lower flammability limit) has to be present but this is the general principle.

The basis is the rate of oxidation.  Think of a piece of wood - put 1/2 in the flower bed for 3 years to rot away and put half in a fire.  Both are the result of oxidation, the later at a faster rate.  Both generate the same amount of heat, one at a rate not discernable to the human touch and the other can cause a skin burn.  

If you are refering to spontaneous combustion from an oily cotton rag, it is slower than an explosion, but faster than the rotting.  The O2 does a little oxidizing which causes some heat.  The little heat causes the air to move and draws in more O2.  This cycle keeps on going until the heat is enought to erupt into flame.  Did you ever put your hand into a pile of grass clippings about 2 days later?  These have erupted into flame, especially hay piles on the farm.  Oily rags have to have 100% air circulation (to dissipate the heat) or zero air circulation to prevent spontaneous combustion.

Now to the serious problem of oil and O2 explosions.  Oxygen is looking for something to unite with and organic materials, such as oil, are its favorite target.  When the O2 is under pressure, such as in cutting torch and hospital rooms, it will penetrate faster and deeper, which means it has more stuff to combine with and will build up heat faster than the stuff can absorb or dissipate.  Thus, just the natural oil from the palm of your hand is enough on a compressed O2 fitting to cause it to explode after you have assembled it and turn on the supply valve.  

The other side of this is when the O2 is not compressed, such as released into a hospital room, but the stuff of the room is super saturated with O2.  Then it only takes a small spark to set it off, such as a cigarette or the spark of a nail in a heel of a shoe across a concrete floor.

Hope this helps with the practical and the chemistry understanding.

The major hazard from oil and oxygen is caused by the absence of the diluent gas nitrogen which is normally present since we often have oil in air. Kersosine lamps are fairly safe. As the oxygen content of the air goes from say 21 mol% to 100 mol%, the autoignition temperature of the oil drops from say 700 deg F to ambient. This is why oxygen cylinder fittings have labels that say, "no oil". There was an accident in a dry dock in the UK. Workers were using oxygen instead of air to drive power tools. The oxygen content of the confined workspace gradually increased until many "safe" materials ignited spontaneously. There were several fatalities.