To clarify France never left NATO, instead they left the NATO military structure.

Basically all NATO members contribute a share of their officers and soldiers to the common NATO forces which are used in operations and missions. These military forces are under the command of the Supreme Allied Commander EURope(SACEUR), who in turn is always an American general. The headquarters of the NATO command structure was the Supreme Allied Headquarters Europe (SHAPE), which was also located in Paris.

France kicked out SHAPE and stopped contributing forces, but was still an active member of NATO, they would intervene in any invasion of other NATO countries and would continue to work with NATO. Today SHAPE is located in Mons, Belgium.

Charles De Gaulle was president at the time and he was a fierce nationalist, believing France to be a world power and always put french interests first. NATO had concentrated a lot of power into the hands of the United States, and other NATO countries had basically become their protectorates, by pulling out of the structure De Gaulle allowed the french government more control over their own nuclear weapons and military forces and made them a more independent actor on the global stage.

There was really never any attempt to please the Soviets, instead it was mostly a move by De Gaulle to allow France more say in international politics, instead of always following the US. In retrospect he was probably right.

Today France has reentered the military structure of NATO under the presidency of Sarkozy. With this reentry France has received some perks, for example the position of Deputy-SACEUR always used to be British and occasionally German, but recently some have been French. The position of SACT (Supreme Allied Commander Transformation) is held by Dennis Mercier, a french air force general.

EDIT: clarifications.