He doesn't eat it, he lets it dry then uses it as fuel for a fire to cook his bread over it. In those times when famine became so severe that wood was scarce, this would be a common practice among the people to survive. So it was an extreme example that God comissioned to Ezekiel to perform as a warning against the Israelites of the coming famine, to show them how they would soon live.
That is why he was only to do it in the sight of everyone, and eat it as he would normal bread, with no look of disgust, to make it look as an everyday thing. Now since outside of the direct command of God this action was a violation of the Jewish law (you're suppose to bury your excrement), and he was raised as he was, Ezekiel couldn't get past that in conscience so God allowed him to use cow manure instead.
God often used prophets to act things out as a symbolic prophesy against Israel because their wicked deeds (Hosea is probably the most extreme example). We don't really understand because they were an ancient, very symbolic/storytelling culture and we are a very modern, factual/scientific culture, but thinking it's weird isn't a good reason to disregard it.