Can a doorbell transformer sit loose inside main panel?
You won't want to install the transformer inside the panel. Instead you'll install it on the panel.


Remove the retaining nut from the threaded fitting on the other side of the transformer (the bit in your hand). 
Feed the wires and threaded fitting through an appropriately sized knockout in the panel.
Thread the wires through the retaining nut, and tighten the nut down on the threaded fitting.
Connect the primary wires from the transformer to the proper terminals in the panel.
Connect the low voltage wires to the terminals on the secondary side of the transformer.