Hi and thank you for using Just answer .. Can you tell me if this oven has a pilot lite for the burners or if it has a electric igniter?
Well, sorry I missed you ... ovens like furnaces have built in safety items. Your problems sounds like you have an ignite-er problem. The built in safety allows the fans to come on to disperse any gar that may be building up. Also, these units have a safety switch on the door, so when you open the door the burners will cut off. These switches do wear out/go bad and it may not be making the connection needed to run the burners. The click sound you heard could be a breaker/safety breaker.
You can also try and reset the system ... turn off the main switch and wait for a couple of minutes then turn it back on. Sometimes these ovens needs to be reset for some reason. A factory cork as it were. I would like to help you out some more, to narrow down the problem. Some answered questions would help. Try what I suggested and a meter would be a good tool to use. If I'm still here when you come back and have questions, please ask. Otherwise another expert will come in to help you out. I wish you the best. Gary