From what I read, when a RHP throws a curveball or a slider, that pitch will curve away from a RH Batter and into a LH Batter. And I also read that a ball curving into a hitter is harder for that hitter to handle than a ball curving away. That would mean that a RHP is more effective against LH Batters than he is with a RH Batter. And the same logic for LHPs. That is, LHPs curveballs/sliders curve into a RH Batter and away from a LH Batter. Therefore an advantage for the LHP over RH Batter.
But I have also read that RHPs have advantages over RH Batters and LHPs with LH Batters.
Traditional thinking, IIRC, is that it is harder hitting a ball tailing AWAY than coming IN. So the defense wants Righty/Righty, Lefty/Lefty, and the offense wants Righty/Left or Lefty/Righty.
Another is the fact that a right-handed batter facing a left-handed pitcher doesn’t have to turn his head as far to see the ball coming out of the pitcher’s hand than with a right-handed batter.
Your last statement is correct; RH pitchers have an advantage over RH batters. LH pitchers have an advantage over LH batters. This tendency is almost universal. 95% of all major league hitters will hit worse against same-handed pitchers. There are lefthanded pitchers in the major leagues who are used almost exclusively to face lefthanded hitters. This is also why switch hitters will (almost) always bat from the opposite side of the plate. When facing a righthanded pitcher, Roberto Alomar will always bat lefthanded.
As is already poinjted out, this is largely because it is EASIER to hit a ball curving towards you, not harder. A pitcher has more room to work with on the outside part of the late if he’s curving balls away from you. It’s harder to percieve the break on a ball away from you, especially if it’s down and away. Snooooopy also points out that you have an easier time seeing a pitch from a pitcher who throws with the opposite hand.
An interesting point that proves the validity of the direction-of-the-curve explanation is screwball pitchers. A screwball is a (difficult to learn) pitch that curves the opposite way from most breaking pitches. Almost all screwball pitchers are lefthanded who learned to throw it so they could use it against righthanded pitchers. But such pitchers often have unusual trouble against LEFThanded hitters, because they’re throwing pitchers that curve towards them like a righthanded pitcher would.
For that reason, you will on very rare occasions see switch hitters hit from the same side of the plate as lefthanded screwball pitchers. I once saw the aforementioned Roberto Alomar do this against noted screwball pitcher Jim Abbott. Alomar hit a home run.