I think it's about the string vibrations. The weight determines how much it will vibrate, in hz (times per second), in tandem with the tightness of the string.

...and I think that's about as far as it gets without googling information.
Randall RG50TC, Roland Cube 15x

LTD MH-100QM, Washburn X12

Effects and Etc.
Ibanez Weeping Demon, Boss FV-500H, Ibanez TS7, Ibanez PH7, dbx 31-band EQ, ISP Decimator