What are BCAAs?
How do BCAAs Increase Muscle Growth?
Leucine, one of the three BCAAs, helps stimulate protein synthesis in your muscles by activating certain pathways within your body, in turn helping your muscles grow. A 2017 study showed that those who had taken BCAAs during their workout experienced a significant increase in protein synthesis compared to a placebo group that did not consume BCAAs. 
How do BCAAs Combat Muscle Soreness?
The soreness you feel after you exercise is called Delayed Onset Muscle Soreness (DOMS), and is caused by small tears in your muscles that occur while working out. Consuming BCAAs has been shown to decrease muscle damage, as a result reducing the length and severity of DOMS.  A 2010 study tested the impact that BCAA’s have on DOMS by giving one group BCAAs before they squat, and another group a placebo. The group who consumed the BCAAs experienced less soreness than the placebo group did. 
How do BCAAs Reduce the Effects of Exercise Fatigue?
Your muscles use BCAAs during workouts, causing the level of BCAAs in your blood to decrease.  Consequently, the levels of another essential amino acid, tryptophan, increase within your brain.  Tryptophan is converted to serotonin which contributes to your level of fatigue.  By fueling your body with BCAAs, it will reduce the amount of serotonin that reaches the brain, therefore reducing the effects of fatigue. A 2011 study showed that those who took BCAAs before their workout had experienced improved metal focus throughout a workout which was a result of reduced fatigue.