What causes the AC compressor to continue running and the freon to continue to circulate after the fan inside the house clicks off?

Main components of AC are compressor & fan motor. The louvers are operated by fan motor which is powered separately. When the louver is off, it doesn't affect compressor work.

- Be-Cool Expert

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possibly a stuck contacter in the outdoor unit. or your indoor fan is overheating and shutting off

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could be many different things. Depends on what system you have. Some of the newer machines are controlled by integrated circuit boards. Its a possibility that your indoor blower could be tripped off on thermal overload but honestly I doubt it. Possible contact could be stuck on compressor. T stat could have a problem but would need to know if the compressor ever turns off after the "fan inside the house clicks off"