The air from the foot control drives pnuematic handpieces (of course) but is also used to activate the coolant to the handpiece (both air and water).
Just as with a toggle used to turn on a secondary valve, the air pressure from the foot control activates a secondary valve to cause the water to run. This air needs to exhaust (relieve) so that your water will then shut off (just like with a relieving toggle). This is why you can hear an audible "hiss" when you take your foot off of the foot control. If the foot control is clogged or sticky, it may not relieve so the water may continue to run.
If this is the case, disassembly and lubrication of your foot control may be in order. I went into more detail on that in a previous post.