The most simple answer is they don’t soundproof a room. As others have said in more technical detail, they stop sound reflecting back, so from inside a room they make it very “dead” (think the opposite of a large empty echoey warehouse or something) so you can only hear what is coming from the speakers. This is important when mixing or mastering recorded music.
Acoustic foam/panels, and certainly more low-budget options like egg boxes do not stop sound leaking out of the room however. To sound-proof a room you need thick dense material and ideally a floating “room within a room” design where the room is totally isolated and all air gaps are separated from the outside of the room. It’s difficult, and expensive, to do well. Simply sticking stuff on the walls is going to make no difference at all.
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