Your power supply can't supply the current that the module takes.
The module, like all GSM modules, takes 1.9 A peak for 580 µs, every 4.8 ms when transmitting.
When there is no SIM, the module does not transmit, and it never finds out that the power supply is weak. When there is a SIM, the module finds a network, and tries to register. As it does so, the power supply dips, and the module turns itself off.
I have had this happen before. I am sorry that I didn't realise what the problem is earlier from your description.
You need a 2 A power supply, or a Li-Ion battery, or a capacitor of about 10,000 µF and a 0.5 A power supply.
What are you using at the moment?