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I read several papers on physical layer security that explain several methods to maximize the secrecy rate. However, I didn't find a very clear definition about the secrecy rate. What is it? How is it computed?

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Here is a pretty clear paper explaining secrecy rate and uses over fading channels.

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.105.2844&rep=rep1&type=pdf

See equation 3 for the formal definition. One way to think about it is like a tug-of-war between the main channel and wiretapping channel. If the main channel has a higher capacity then the secrecy rate is positive, otherwise it is negative.

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