Why do jet engines sound louder on the ground than inside the aircraft?

Sound is a pressure & velocity wave in fluid medium, i.e. air. Air molecules wiggle back and forth and bump into other air molecules so they wiggle too so you have a whole chain of wiggling air molecules.

The jet engine moves air molecules A LOT, hence it's extremely loud. As the sound moves away from the jet engine the energy disperses over a larger and larger area and so the sound pressure level drops. The pressure drops by half every time you double the distance. That's 6 dB per doubling of distance or 20 dB per decade. If it's 120 dB at 10 meters, it's still 100 dB at 100m, 80 dB at 1km and 60 dB at 10km. That's why you can easily hear it on the ground.

There is no easy way for sound to get into the cabin, because the cabin is air tight and fully sealed. The air molecules outside can wiggle like crazy but the air molecules inside don't care.

It's still fairly loud in the cabin but that's due to mechanical sound transmission through the wings and the fuselage. The vibration of the jet engine wiggles the wings which will wiggle the fuselage which will wiggle the panels which will wiggle the air molecules inside the cabin, which will wiggle your ear drum. Planes are carefully designed to minimize the transmission but the amount of energy from the jet engine is enormous, so even if you eliminate 99.999% of the energy, it's still quite loud and getting to 99.9999% or 99.99999% is difficult and very expensive.

First, the cabin is quieter because the fuselage walls are designed to limit the transmission of sound from the engines.

Second, on most commercial aircraft, the engines are suspended beneath the wings, which block the noise from the engines before it can strike the fuselage.

There is a howling 500 - 600 mph gale blowing outside the aircraft, the like of which you have never experienced on the ground, and this carries away most of the sound before it can enter the fuselage. Some sound is also reflected by the fuselage. A few passenger aircraft like the Caravelle, VC10 and Tristar have engines at the rear, which also helps to reduce sound levels in the cabin.