You need to add a third "something else" class to your network. There are several ways you can go about it. In general, if you have a class that you want to detect you should have examples for that class, so you could add images without cats or dogs to your training data labelled with the new class. However, this is a bit tricky, because the new class is, by definition, everything in the universe but dogs and cats, so you cannot possibly expect to have enough data to train for it. In practice, though, if you have enough examples the network will probably learn that the third class is triggered whenever the first two are not.
Another option that I have used in the past is to model the "default" class slightly different from the regular ones. So, instead of trying to actually learn what is a "not cat or dog" image, you can just explicitly say that it is just whatever does not activates the cat or dog neurons. I did this by replacing the last layer from softmax to a sigmoids (so the loss would be sigmoid cross-entropy instead of softmax cross-entropy, and the output would not be a categorical probability distribution anymore, but honestly it didn't make much difference performance-wise in my case), then express the "default" class as 1 minus the maximum activation value from every other class. So, if no class had an activation of 0.5 of greater (i.e. 50% estimated probability of being that class), the "default" class would be the highest scoring one. You can explore this an other similar schemes.
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