Finding a script
Assuming that the play you are wishing to produce is new and is being shown for the first time then the most common way for a script to be found is either through the writer or the producer. The writer is usually trying to find a way to get their script produced and will be looking for someone to help them get it off the ground. Conversely a producer may be looking for a script to help bring to life for various reasons. There is usually some financial motivation, but there may be other reasons as well. A good producer will recognize a good story and want to help get it off the ground.
This is the very first part of how a play comes to life. Having a writer and producer meet who want to help get the story off the ground is where it starts. It will be helpful for the two to have a good relationship and be able to work well together.
If this is not the first time the play is being shown. Then usually the process will be the same except with the producer and the director meeting instead. In this instance the director will want to put on a production of the show and will need the help of a producer to get it off the ground. There will also usually be a royalty fee that must be paid.