What is the significance of flies?

At the start of the episode, a fly walks across Dolores' face, and then over her eye. This shows that she doesn't react like a normal human would, even though she looks human.

Later, she is questioned in the lab: "Have you ever hurt a living being?" And her answer is "No".

At the end, when the fly is on her neck, she kills it. She does not show any remorse. However, this is the first time in her decades of existence that she has hurt a living being. So it shows that she is "evolving", and it shows that her evolution may be dangerous, since she has no reaction to hurting/killing living things now.

In addition, the flies appear to represent malfunctions, or evolution, of the hosts (robots). When one of the male hosts has a meltdown while leading the married couple through the mountains on horseback, a fly is walking on his face. That is when he starts having his facial tics and stuttering.

I believe that this is a reference to the common expression "wouldn't hurt a fly". This phrase means that a person is inherently gentle, peaceful and non-violent. Such a person would do nothing to ever bring any one or any thing to harm.

A running theme in the first episode deals with the safety of the Westworld theme park and the nature of its artificial intelligence inhabitants (the hosts).

When a fly is seen moving across the face of a host, the AI does nothing (not killing it or even trying to shoo it away); showing that it is not programmed to hurt or even bother with such a lifeform.

It also helps the audience to distinguish hosts from guests (humans) because people are actually very likely to shoo such pests away.

During the final moments of the episode, Dolores is interviewed by Westworld staff for signs of abnormality. The interview goes well and Dolores doesn't seem to be malfunctioning like other hosts. Dolores is sent back to the theme park to resume her routine.

In the final scene Dolores does something extremely "out of character" by killing a fly. This element acts as a cliffhanger as it shows that something clearly is different about Dolores.

The flies are being used for multiple storytelling purposes.

The most important, obviously, is to show Dolores's evolution away from what we'd consider to be First Law programming: that she's programmed not to hurt any living being at the beginning, even to the point of letting one crawl across her eyeball (which also emphasizes that she's not human), to her swatting one and killing it at the end of the episode, indicating a change in her programming/consciousness.

Secondly, they are part of the huge contrast between the Delos corporate behind-the-scenes world which is black and white and sharply geometric in design, and antiseptic in comparison with the all-too-organic make believe Westworld park.

There is also the psychological idea of rot, disease, and the fact that the Beelzebub is sometimes translated as the "Lord of Flies", bringing in the idea from Episode 2 ("Chestnut") where Ford mentions that you can't play god without being acquainted with the devil.

And, most specifically, in the second episode, when it's discovered that Maeve is suffering from an MRSA infection (which indicates that the hosts might be comprised of biological components as well as hardware), one of the techs proclaims, "No wonder we have a fucking fly problem!" Which gives us further clues into the nature of the hosts, the poor hygiene/maintenance of the park and guests for the hosts, as well as making the prominence of the flies motivated by events, and not just a stylistic choice.