Why is the temperature inside a terrarium and in the air different?

To understand this fully you would need to now a bit of quantum mechanics - specifically the photo electric effect for which Albert Einstein won the Nobel prize. Briefly put the higher frequency end of the electromagnetic spectrum which occurs in natural sunlight can penetrate the glass as with a greenhouse and heat the contents. The energy that is re-emitted from the warmed contents is at a lower frequency that is less penetrating and the result is a net heat gain. There are limiting factors in that there are various ways in which the terrarium can loose heat nevertheless as with the greenhouse it serves it's purpose.