Why are large speaker cones required to to produce loud low frequency sounds?

To radiate sound effectively, you require the radiating object to present a good impedance match to the air surrounding it. In so doing, you will maximize the radiation resistance of the sound source. In practical terms this means the physical dimensions of the object should be approximately the same as the wavelength of the radiated sound. For example, to radiate well at 1000Hz (wavelength ~1 foot), the radiator should be about 1 foot in size- so a 12" loudspeaker will be well-coupled to the air surrounding it at 1000Hz; a 6" speaker will be well-coupled at 2000Hz, 3" at 4000Hz, 1.5" at 8000Hz, and so on. This is why the tweeters in your hi-fi speaker system have a cone diameter of about 1", the midrange will be 4", and the bass will be 12"- or as big as you can fit into the enclosure of your choice.

This rule extends to low frequencies too, but here the practicality of the scaling law breaks down: To radiate well at 100Hz requires a speaker 10 feet in diameter, and a pair of these will not fit in your living room. Instead, you make up for the poor impedance match with more cone excursion: You drive the speaker cone farther back and forth so as to pump the same volume of air that the big speaker would, only with a smaller speaker.

This is an inefficient way to get good low-frequency response, and requires the power amp that is driving the woofer cone to feed a lot more juice to it- but at the present time, power amplification is inexpensive compared to the cost of (for example) an 18" diameter loudspeaker, which needs to be mounted in an enclosure the size of a refrigerator in order to engage the air properly.