The phrase "look me in the eye" is used to test how truthful someone is being, usually someone close to you. The idea is to shame them into telling the truth, because looking a loved one or somebody you deeply care about in the eye and deliberately deceiving them when they're demanding you tell the truth is tantamount to betrayal.

In some sense this is literal: it means "look at me directly, and do not look away". But the whole sentence is an idiom: you say it when you suspect that somebody is lying, and you want them to look at you while they speak. The idea is that if they respect or love you, they will not be able to lie to you while looking at you.

looking at someone without showing fear, weakness or embarassment