As others have already noted, it could mean several things from a shyness to look at someone directly, a disregard for someone (as if to say you are so unimportant to me I don't even see you) or just cultural manners most people were taught as not to stare prolonged at someone as it comes across as hostile or creepy. It of course could also mean someone is lying to you if they are also telling you something as unless they are good at it, most people can't look at someone in the face when they do it.
For me being a lifelong introvert, I was never 100% comfortable looking at others directly and was always looking down that had nothing to do with my actual feelings towards that person or that I was trying to communicate some hidden hostility or judgment AKA "micro-aggressions" (that I consider a made-up term). It could also be the fact, that for me in certain areas and environments that I grew up in, people were taught to basically disregard others unless necessary or "look through them" as a survival mechanism.
Combined with my shyness, I found this was a hard habit to break when I worked at in an occupation that required the exact opposite method of purposefully staring down others so they know you are watching and alert so to give off a "don't even think about fucking with me" vibe.
So maybe it is simply any of the above or it might be that the person really doesn't want to engage with you for whatever reason and trying to give a non-verbal hint. I would suggest you take the hint and move on. Unless this person is a friend, significant other or family member whose working things out is important to you, who cares what their motivation is; just leave them alone.