Jake Blocker's excellent answer to How does one dogleg from Florida to a sun-synchronous orbit? explains what a dogleg maneuver is, what it's for, what it looks like, and even what a real dog's leg looks like for comparison!
A dogleg maneuver is done to change the inclination of a certain payload, and the reason it limits the payload capacity is most likely due to the cosine losses.
In my answer to it I mention an incident which might be the reason that doglegging to orbit was invented, but I don't know. I also describe it in this answer to History of multiple-payload launches?
Question: Which launch was the first to dogleg?