Please explain real-world procedures behind this "ATC joke"
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.