Hitters tend to perform better against opposite-handed pitchers than they do against same-handed pitchers. This effect is mostly due to the ability to see the ball better out of the pitcher's hand.
Right-handed batters average about a 7% better OPS versus left-handed pitchers than they do right-handed pitchers. Left-handed batters average about a 13% better OPS versus opposite-handed pitchers.
So how has usage changed since 1960?
The difference in percent total LHP and percent of PAs LHB see vs. LHP is growing closer. The % total LHP is now lower than the % PA all batters have seen vs. LHP. Now, left-handed pitchers are beginning to see more PA vs. RHB. However, the amount of PA for LHB vs. LHP has also increased. This seems to show that in the early 90's to mid 2000's, there were more LOOGYs (Left-Handed One Out Guys) and that either there are more left-handed starting pitchers or more left-handed relievers are facing both types of batters.
I also looked at how overall OPS has changed over time as batters face a different percentage of left-handed pitchers.
Here you can see the crazy drop-off in how often right handed hitters faced left-handed pitchers. It's also odd that this is correlated with the high-offense eras of the 1990s and 2000s. There was a small period where right-handed hitters were hitting left-handed pitchers much better, but split has remained relatively consistent.
Left-handed pitcher usage has been steadily rising versus left-handed hitters since the late 1990's, but seems to have hit an equilibrium point for now. The OPS split also appears to be getting larger, as the OPS vs. LHP is decreasing more rapidly than the OPS vs. RHP.
Starters are defined from baseball-reference as having started more than 60% of the games in which they appeared. Relief pitchers are defined as having relieved in more than 80% of the games in which they appeared.
The National League will tend to have a larger percentage of pitchers, since there are more teams in the National League. The percentage of right-handed relief pitchers has been increasing since the 1970s. The percentage of right-handed starting pitchers has been decreasing in the American League since the 1970s as well. However, the percentage of left-handed relief pitchers has been fairly steady since the 1960s.
Managers have definitely been playing the matchups differently. However, it does not appear that all of this is due to LOOGY usage, since right-handed hitters have been facing more left-handed pitchers in recent years as well.