The largest ring will begin to vibrate strongly first, followed by the second largest, and so on. The smallest ring starts to vibrate at the highest frequencies. Keep shaking the cardboard faster and faster, and notice that the largest ring will begin to vibrate strongly again. Each ring will vibrate at more than one frequency, but the shape of each ring will be different for each resonant frequency. The rings will also have different resonant frequencies if you shake the board up and down instead of sideways.
Stiffer objects have higher resonant frequencies, whereas more massive ones have lower frequencies.
The biggest ring has the largest mass and the least stiffness, so it has the lowest resonant frequency. Put another way, the largest ring takes more time than the smaller rings to respond to an accelerating force. As you shake the sheet, the BB will tap against the ends of the straw at the same frequency as your vibration.