Yes Liquid Plumber will cling to the hair in the drain and eventually dissolve the hair in the drain causing the drain to unclog but it may take 30 minutes to an hour for it to work depending on how clogged the drain is.
You can also pour Liquid Plumber down the drain often as in a few times per month to keep the hair from clogging up the drain before it gets to the point of clogging.
You can also remove and dissolve hair from a drain using baking soda and vinegar.
Pour a cup of baking soda down the drain and then pour a cup or two of vinegar down the drain and the baking soda and vinegar will react to each other and after around 30 minutes the hair should be dissolved.
Cover the drain with the drain plug while you let the vinegar and baking soda work in the drain.
Then pour some hot boiling water down the drain to help flush out the hair and baking soda and vinegar.
This will also help get rid of any odors in the drain as well.