Well, eyes to live for actually.
So, what on Earth is an insect doing with such obviously mammalian 'eyes' on it wings, albeit eyes that can't see? You would think a tasty morsel like that would want to keep hidden and not have bright colours advertising it, wouldn't you? They surely can't be there for us to admire, can they?
But of course, it has nothing to do with us. It has to do with birds!
You see, to find a mate, the butterfly needs to advertise itself, and reproducing is what it is for. But it needs to avoid getting eaten too, so we have a classic tension leading to an equilibrium where finding a mate needs to be balanced against the need to avoid being eaten - at least until after it's found a mate and produced eggs for the next generation.
Enter our predatory bird.
This bird needs to find food for itself and it's chicks and it also needs to avoid being eaten. So again we have another equilibrium resulting from the tension between finding food and being exposed to predators.
Let's see things from a bird's eye view.
An upside down butterfly.
But cover the fore wings and what do you see?
You see, the birds who survived were those who didn't stop to think, "Oh! What lovely eyes!", because those would be eyes to die for, but those who were up and away before they even knew what their wings were doing. Those who did it best left more descendants, so a rapid flight response to a mammalian eye evolved in the presence of mammalian predators.
Now what has this got to do with the Peacock Butterfly? We'll, birds with these super-fast reflexes were in their environment and were trying to eat them, so, whilst the butterfly was needing to evolve ways of attracting a mate without being eaten, it happened upon a pattern which looked a bit like the eyes of a fox or a stoat, or some other predatory mammal, which is what the bird's genes needed not to be eaten by.
To begin with, the marks might not have looked very much like mammal eyes just so long as in poor lighting or for birds with poor eyesight, they triggered the flight reflex. As the survivors with these slightly eye-like markings became more numerous it was those on whom natural selection worked, so the eye marks became more eye-like and better at attracting butterfly mates.
Also, the butterflies who carried genes which meant they found the eye-like markings attractive would have left more descendants because those genes would have benefited from the presence of genes for making the eye-like markings, so sex-selection would have reinforced this evolutionary process.
From the bird's genes point of view, it was a better trade off to miss a snack than to be eaten. From the butterfly's genes 'point of view' (it's a metaphor, don't get over-excited) it was better to avoid being eaten and to find a mate. Win:win for the butterfly genes and a powerful driver in favour of greater and greater perfection. But perfection not in looking exactly like mammalian eyes but in triggering the flight reflex in the bird. So, maybe two sets of eyes worked better than one, and additional marks which make it look like the eyes are coming closer were even better.
All it has to do is open it's wings and flash those eyes, and the same mechanism works with a prospective mate too!
So, we have insects with mammalian eye-markings because their predators were preyed on by mammals.
And all the butterfly was trying to do was find a mate and avoid getting eaten. Now, what intelligent designer would have have designed that, unless it was one who loved watching birds get the surprise of their lives and denied a meal with a trick?