I'll try to help you understand if I can.

For the gambler's fallacy, many people believe that subsequent events are related, when in reality they are not. Take a die. A person who rolls 10 times and does not get a 6 might think that getting a 6 is more likely b/c of the 10 previous rolls. While rolling rolling 11 times and not getting a 6 for all 11 rolls is unlikely, your chances of rolling a 6 given that you've rolled 10 times is still 1/6 (the same as if you only rolled once). In the first scenario, you're considering an overall probability whereas in the second scenario, you're considering a conditional probability.

For regression towards the mean, it's better if we approach this from a frequentist's POV. Frequentist statisticians (also known as classical statisticians) believe that in a process where the variables observed are iid (independent and identically distributed), there are fixed parameters that can be estimated. The mean is one such parameter. This fixed value doesn't change. So connecting this reasoning to your example, the batter has a (unknown) "true" batting average which we can try to estimate. Hypothetically, lets say that we do know what his "true" batting average is and that each time he goes up to bat is independent. Then if one game, he has a killer day and scores home runs on every bat, you'll have observed an event that is way above his true batting average. The probability of this is extremely rare and future observations will likely be lower (i.e. closer to the mean). You're not in any way assuming that the observations are dependent. You're just observing a value that is rare. Values that are closer to the mean are more likely to happen and therefore the next observation you see will be closer to the mean.

Hope that helped.