Trackpad on MBA keeps automatically clicking
This sounds like either a faulty trackpad or possibly a swollen battery.  I know the battery suggestion sounds strange but the reasoning is that as a battery swells it puts pressure on the components behind it, which for most of Apple's portables includes the trackpad.  Basically it could be slightly pushing on the back of the trackpad at all times, causing even minor touches to register as a click.

The only way to check for this is to remove the bottom case, which you may or may not be comfortable doing.  iFixit has a guide for replacing the battery that shows the steps to remove the bottom case and provides a reference for what a healthy battery should look like.  If the battery is not swollen then the other likely hardware fault is the trackpad, which isn't common but isn't all that uncommon either.

Before you tear into your Mac though you can at least rule out software as the culprit.  You can boot to your recovery partition by performing the following:


Power down your Mac normally ( > Shutdown)
Hold option while you power on your Mac
You should see a list of hard drives, one of which is likely 10.9 Recovery, select that and press return.  If you're instead presented with a password request you will first need to enter your firmware password before seeing the list of drives.


Once you're booted to the recovery partition you can test your trackpad but clicking/not clicking on items.  If the issue persists then it's likely one of the hardware faults I outlined above.

Hope that helps!