Suppose you're travelling with a bearing of 45 degrees to North. We call this direction "Northeast". Are you travelling North? Some might say yes. Are you travelling East? Some might say yes. So are you travelling both North and East at the same time? Some might say yes.
So your travel vector is a superposition of the North and East vectors so we might say you are travelling both North and East at the same time.
It's not so strange to call the superposition vector the cat being both dead and alive at the same time. The strange thing, in my opinion, is that physical states are represented by state vectors in a Hilbert space. But once we accept that physical states ARE represented by state vectors in Hilbert space, it's not so strange to call superpositions "both at the same time".
Though I do agree with comments that this feature of quantum mechanics starts to stretch the philosophical boundaries of human thought and language.