Headphones Won't Plug Into Macbook Air
If your headphone jack cannot go in, there are only three reasons why this will occur:

The port is obstructed
The headphone jack is the wrong size
There's a manufacturing defect of the MacBook port

You can't remove dust or dirt if there's no dust or dirt to remove.  Also, you don't want to use a tooth pick as that will push objects in further and can even damage the port.  Compressed air is a better option and if you can do it from the back of the connector (requires opening the case), that would be the most effective method.
Forcing a connector is highly discouraged; if you can't make the connection with the reasonable amount of force that's required, stressing it could break things.  If it doesn't go in, something is wrong and excessive force will make it worse.
Verify that the headphone jack can be inserted into other devices and that you are indeed using a 3.5mm jack.  Finaly, try other headphones.  If  none can fit, then it's very likely you have a Mac with a manufacturing defect.  If that's the case, try contacting Apple Support. You may have some luck with them swapping out the case even if you're outside the warranty period.