Please explain real-world procedures behind this "ATC joke" One of the well-known kind of the "ATC jokes" (funny excerpts from real or invented communication between pilots and air traffic controllers) has recently caught my eye.

I can't seem to understand what actual procedures or rules does it refer (or allude) to:


  Pilot: "Approach, Federated 303's with at 8000' for vectors ILS, full stop.
  
  Approach: "Unable Federated 303. The ILS is out of service."
  
  Pilot: "We'll take the VOR then."
  
  Approach: "Sir, the VOR's in alarm right now. Standby."
  
  Pilot: "OK, guess it'll have to be the ADF then."
  
  Approach: "303, unable the ADF right now for traffic saturation."
  
  Pilot: "OK, approach. State my intentions."


(src)

Could you please enlighten me on what does the last (highlighted) line is intended to mean and/or convey? I understand it's a word play on "state your intentions", but what exactly is it supposed to mean?
As you commented, it's a play on "state [say] your intentions". However, the pilot has said his intentions several times and ATC rejected them each time, so he's turning it around by asking ATC to tell him what intentions won't be rejected.

The "intentions" in this case are the type of Instrument Approach Procedure (IAP) he wants to use. Typically there are several for each runway, for use by aircraft with different levels of equipment; ILS, VOR and ADF (actually NDB) are common options in order of decreasing desirability.

Depending on the weather, there may be two additional IAPs available at any airport which aren't usually charted: visual and contact. There's a possible second-level joke here in that ATC isn't allowed to offer a contact approach, but the pilot is allowed to request it--so if the pilot is asking ATC to tell him his intentions, then in the logic of the joke, ATC could tell him he's requesting that. OTOH, if the weather were such that a visual approach were available, ATC probably would have assigned it to him already without any requests.