What does "Transient Response Recovery" means in the output specification of a AC-DC power supply?
Transient or step response means how quickly a power supply reacts to changes in load. In your case it may or may not matter depending what you're driving with that 5V. If you've got something like a motor or other power electronics, the load can change suddely. 



With regards to the tantalum packing it in, it's most likely because you're simply overstressing it. I can't tell from the image the part number but if it's an SMPS, you can have significant input ripple current depending on load.