Origin
Many still argue about the origin of this phrase. There are several different theories as to where it first came from. Let's look at some of them:
What we do know is the earliest known use of this saying is from a title of a play in 1957. There is even some evidence that shows theatrical memoirs and personal letters from 1920 that may be related to the phrase. As for the theories...
The Superstitious Theory: For some superstitions, wishing good luck for someone can cause bad luck, and wishing bad luck on someone may cause good luck. Therefore, telling someone to have perceived bad luck of breaking their legs would result in them having good luck.
The Roman Origin Theory: In the Ancient time of Rome, a popular form of entertainment were colosseums, where gladiators would would compete and fight to the death. The people who were watching would yell "break a leg" to the gladiators as a form of good luck, basically telling them to stay alive and to merely disable the opponent by breaking his leg.
The Most Common Theory: This theory says the phrase originated from tradition in Britain. In the time of Shakespeare's King's Men, sometimes the actors would receive tips from the audience depending on how good the performance was. A good performance was rewarded with tips of money, while the bad was a bunch of rotten vegatables thrown at you. For the good performances, actors would 'take a knee', breaking the sight of their leg line, and collect the money thrown to them. So wishing for someone to "break a leg" is telling them you hope their performance is a success, so they might kneel down at the end and collect their tips.
Examples of The Phrase Being Used
"Hey, break a leg out there, Janice."
"I hope you break a leg on the upcoming math test, man."
"I wished my friend to break a leg when he went to audition for the part of Julius Caeser."
Note: For most old phrases, sayings, and idioms, finding the exact person or date in which a phrase was originated is impossible. What's provided is the earliest known date the phrase or saying was being used in.
Usually the phrase is a quote from an old newspaper, book, poem, play, etc. If it is already being used by a newspaper though, then the saying must already be common knowledge among the people of that time. This means the origin of the saying is older.
For instance, if someone or something from 1660 is being quoted using a particular phrase or saying, you can assume that the saying must be older, as it's already popular and being commonly used in 1660. How far back varies from phrase to phrase.
There are some cases though where determining the origin from an exact person is possible, however these are not common. In short, most of the origins for these sayings give you a date so you can get a rough idea on when these phrases were being used.