This thread is getting old but this LED&KEY device seems to still be fairly popular so this may still help someone else. I'm using one to control a radio receiver and the constant buzz can get pretty loud.
I found this article (A second look at the TM1638 LED & key controller) which mentions the same issue which was solved by inserting a 100Ω resistor between VCC and the VCC pin. I also added a 1000μF electrolytic capacitor across the VCC and GND pins (well, fairly close to them anyway). This quieted down the buzz a lot though it's still there. The resistor is key though the capacitor helps a bit.
I also tried moving the breadboard connections to a separate, smaller breadboard, away from the MCU and radio receiver chip and was able to use lower values (10Ω and 47μF). I was no longer able to hear the buzz at all.