I'm a little late to this question, but want to add a little for people who aren't familiar with an aircraft heading.
As you said, the dialog is intentionally silly (and not entirely accurate) but most of what they are saying is loosely based on reality.
The short version is that a vector is where an air traffic controller is talking to an aircraft that is visible on radar, they know where they want the aircraft to go, and they assign a direction for them to fly in order to get there. This gets the aircraft where they want them without the aircraft needing to navigate on their own, and often allows them to take a "shortcut".
The longer version:
A vector as defined in the Pilot Controller Glossary is:
VECTOR - A heading issued to an aircraft to provide
navigational guidance by radar.
(See ICAO term RADAR VECTORING.)
Of course, that leads to:
RADAR VECTORING [ICAO] - Provision of navigational guidance
to aircraft in the form of specific headings, based on the use of
Which then leads to:
HEADING - ...
Wait a minute... They don't define it! Okay, for those people that aren't pilots, a heading is the direction that an aircraft is flying. They are three digit numbers corresponding to the 360 degrees in a circle wrapped around a compass rose:
While technically a heading is to the nearest single degree, in most cases ATC will only assign headings that are multiples of ten, or on occasion five, degrees. So, after North, the commonly-assigned headings would be: 005, 010, 015, 020, etc. all the way around until you get to 355 and then 360. After that it wraps around again.
So if air traffic control wants you to fly North East, they will tell you to fly a heading of 045, South would be 180, and North would be 360 (fun fact, 000 is not a valid heading).
When ATC assigns a heading for a vector, they always have to be cognizant of what happens if the radio in the aircraft fails. Obviously they wouldn't expect you to fly straight ahead in that direction forever, so they include the end result of the vector in the clearance. Usually they also include the direction of the turn for those pilots that aren't very good determining directions. ;-) For example, they could say:
N1234, Turn left, fly heading 345, radar vectors to Fort Lauderdale International Airport.
In this case, they expect the aircraft to start a left turn, roll out on a heading of 345 and continue to fly straight until they get another clearance. If the radio failed, they would turn direct to the airport.
Sometimes they want the aircraft to fly a heading until they intercept a published portion of an instrument procedure. In this case, they would say something like:
N1234, fly heading 045 to intercept <the procedure>
The airplane would then fly the assigned heading until they get to the procedure and then fly the published procedure!