These statues generally pre-date written documents, and the written documentation we do have does not talk about the figural depiction of votive statues, so we are left to interpret the significance of the shape without much guidance.

I am an historian (text person), not an art historian (object person), and an art historian will probably give you a more interesting response. I have heard several hypotheses about the shape of the eyes, but to my knowledge, there is nothing resembling a consensus on this issue. It is fairly common to hear scholars speak of the eyes as expressing a “visage of ecstatic piety” or “diligent prayer.” Ultimately, these are just guesses however.

It is very interesting to note the graphic similarity to the Tell Brak “eye-idols” from northern Mesopotamia. Although the eye-idols date centuries earlier, the dominance and centrality of the eye motif on these figures, and the prominence of eyes on the subsequent Early Dynastic votive statues is likely to be more than just a coincidence.

For more info on the statues in general, check out this introduction: http://www.metmuseum.org/toah/hd/edys/hd_edys.htm

or for a deeper engagement, the standard introduction is:

Henri Frankfort, The Art and Architecture of the Ancient Orient (1970–1996)

Some context: These sculptures were found at the "Square Temple" of Tel Asmar buried in a pit as a method of ritual disposal. Their original context is somewhat confused; we think some of them were placed in the sanctuary as stand-ins for worshippers who could not enter the sanctuary but there's evidence that other sculptures were placed in doors and courtyards-that is, publicly accessible areas. As WedgeHead makes clear there are limits to what we can say about these with no contemporary texts but we can make some more general contextual inferences with the help of later texts. We do know that the act of seeing was often associated with observing marvellous things and attentiveness, and we can say reasonably that a wide gaze facilitates certain kinds of direct relationships between viewer and image. As such, wide eyes work both in a sanctuary context where the image is presented as being attentive before the god and in a more accessible context where the image is presented as interacting with a viewer who is venerating it as a memorial image. In both cases it is the eyes that allow the statue to observe the divinity in the temple or the honor given to it. It is also helpful to note parallels with later sculptures such as the Gudea sculptures which have similarly exaggerated eyes and which we know from inscriptions were carved for temples and so Gudea and his memory might be venerated. If you'd like more information than this summary, you might want to read:

Evans, J.M. The Lives of Sumerian Sculpture: An Archaeology of The Early Dynastic Temple(Cambridge: Cambridge University Press 2012)

Irene Winter, "The Eyes Have It: Votive Statuary, Gilgamesh's Axe, and Cathected Viewing in The Ancient Near East" in On Art in The Ancient Near East. vol. 2: From the Third Millennium On(Leiden: Brill, 2010).