He uses his anchors in a variety of ways during combat. At first, he'll swing them around like clubs and try to hit you with them but he can also whip them at you long range similarly to the blades from God of War. He'll swing them both over head and then bring them crashing down on top of you. If he happens to hit you with them, he'll pull them back to him and you'll come with them. Sometimes it will put you right in front of him while other times you will simply fly in his general direction and go over his head.
There's no easy way of defeating him except by hitting him in the body. The best method is to use powerful bursts of damage like from a shotgun, sniper rifle, or rocket launcher. He can block shots by crossing his anchors in front of him but he won't do this until after you've hit him a few times. For this reason, you want to do as much damage as quickly as possible before he can bring his guard up. Then you wait for him to lower his guard and do it again. Repeat this process until he goes down.
In a one on one fight, an Anchorman isn't terribly difficult (though still tough enough you should be on your guard). He becomes tricky when paired with other enemies. When part of a group, he can toss you around a lot, making it difficult to fight and when paired with other Anchormen, this throwing about can happen multiple times in a row, making it that much more disorienting and giving you little time to readjust before you need to dodge another throw. Try to get as far away as possible or put a large piece of cover between you that will prevent you from being thrown to counter this.