When a team is looking for pitchers, especially starting pitchers, they are looking for the ability to get both left and right handed hitters out. There have been plenty of LOOGY/ROOGY's who have had long careers, but their value to the team is not impactful, due to lack of innings. There are two major factors which usually show the divide in results against lefties and righties: repertoire and ballpark factors.
A pitcher's repertoire is the biggest reason for a difference in platoon performance. There are two types of pitches that usually get hitters out: hard in and soft away. Physically, it makes sense. Let's take a same side matchup as an example. It's harder to get the barrel on a 2-seamer inside than a cutter away (same speed), since you have to get the bat through the zone quicker to pull the ball. On offspeed pitches, once the bat is through the zone, it starts coming back towards your body, making inside pitches easier to hit. This is why sliders are better offspeed pitches for same-side pitchers.
Here's a list of pitches and where the platoon advantage is gained for pitchers:
Same side: 2-seamers, sliders
Opposite side: cutters, changeups/splitters
Neutral: 4-seamers, curveballs
Ballpark factors also influence the results of platoon splits. A left-handed pitcher in Seattle is expected to give up less HR than a righty, due to the enormous left field dimensions there. This does not represent a difference in performance. If the lefty gets traded to Boston, then he will most likely give up more HR, due to the righty-friendly dimensions of Fenway. A great site for these platoon ballpark factors is statcorner.com. I'm not big into the K and BB factors, but they have double, triple, and HR factors, plus an overall wOBA factor. They also have the same factors for minor league parks, all the way down to A level.