When you’re dealing with any mechanical device that has moving parts, those parts will naturally move more freely when they have been loosened up by use. In speakers, both the surround (the ring of rubber or foam that connects the edge of the cone to the speaker basket, and the spider (which centers the moving parts of the speaker) move, and will move more freely after they have had audio pumped through them for an extended period. A good rule of thumb is that playing any sort of full-range recorded music through a speaker at a mid-volume level for around 100 hours is enough to break the speaker in. The hours do not need to be consecutive, so you can play music through a speaker in another room when you are asleep at night, or just use the speaker normally and it will be broken in after about 100 hours of use.
How important is breaking in a speaker? The sonic effects of breaking in a speaker are sometimes very subtle (a more consistent sound) and sometimes more obvious (a softer midrange and warmer highs) depending on the speaker.
The bottom line is that it’s gonna happen eventually anyway; the question is how quickly do you want/need it to happen? It’s very easy to do if you’d like the benefits of a broken-in speaker immediately, but if the speaker is doing the job for you, it’s not something you have to worry about.