In this answer I noted reasons why an optical instrument (that happens to be part of a living creature) would want a large aperture. That made me think about slit pupils at the time, though I did not mention it in that post.
Depth of Field
Do you know how auto-focus works on dSLR cameras, and the “split prism” people used before that? The phase matching works by taking light from each side of the aperture separately and comparing the separate images. This only works if the aperture is in fact large enough to have separated sides.
Now arguably a smaller aperture provides a sharper image already. But the wider aperture allows for accurate ranging: the determination of distance to an object, rather than just seeing it sharply.
This is the effect aluded to by the articles linked by Mołot’s comments.
Now consider this question on the Photography SE: What's the difference between cross-type autofocus points and regular ones?
cross-type sensor is simply two linear sensors crossing each other (making a cross shape), which is superior because it can work with both vertical and horizontal light patterns.
A cross-shaped pupil allows for the angular resolution of a large aperture, while masking off the excess light from the large area of such an opening.
Cameras find it advantageous to use cross shapes rather than slits, so they can focus on a larger variety of objects. An animal may find the same advantage.
Note that the principle is the same regardless of whether the light from each edge is sensed separately. If separate sensing is used, the features can be matched and this is called phase. For a single image taken with an extended aperture, you simply judge the sharpness of the image. In either case, we exploit the physics responsible for reducing the depth of field as we increase the physical size of the aperture.
Diffraction Limited Optics
In the case of an instrument working at full aperture to get angular resolution, the implication is that a larger aperture can produce a sharper image (of the in-focus item). In the case of synthetic aperture instruments the cross-shape ought to provide the additional information. However, I don’t know if a simple projection onto a retina would be sharper or not: the blurring due to diffraction might be caused by the narrow parts and having wide parts as well doesn’t un-blur it.
But you could imagine that the optical system behind that alien aperture is not like a mammal’s. It could be something designed to indeed have a high angular resolution needing a mimimum physical aperture size.
From Mołot’s links, we see that some animals have various special features in the eyes, like multiple fovia or some kind of separate zones for different colors. The shape of the pupil might not make sense if you assume that the eye is otherwise just like ours, but is part of a complete system.