I learned from a book by Brendan Brazier that, when post recovery, when muscles are inflamed, they need blood to shuttle nutrients and take away wastes (like lactic acid). Anti -inflammatory diets will decrease the stress of the body thereby allowing recovery to take place faster. Faster recovery means you can work out again sooner, putting in more workouts in the same amount of time. Thus, you can get stronger than you would if you had not eaten an anti-inflammatory diet.
His was a vegan approach, but anti-inflammatory means what it means. I'm sure that, depending on how meat is prepared, it can be relatively more or less inflammation causing. Eg highly deeply fried meats = highly inflammatory