Tefilla (prayer) is called 'Avoda Shebalev' -- literally, "work that is in the heart."
Good acting is achieved when we find in our hearts what needs to be brought out for the role, then access it and express it at the appropriate time.
By working to develop our acting skills, we come to know, from experience, the difference between our own authentic expression and just going through the motions.
How does this apply to Tefilla??
It can help us become more proficient at being authentic, and allow us to become more aware of when we're just reciting the words (or "sleep-davening!"). Then we can more easily avoid falling into rote patterns, and be better equipped to maintain a real, living, deep, and dynamic relationship with Hashem.
Essentially, good acting requires channeling one's emotions in the appropriate way at the appropriate time, and this skill is paramount to successful Avodas Hashem. There are many sources, both ancient and contemporary, that reflect this idea. (See "Selected Mekoros/Sources" page for some examples.)
Thus, by working to improve and hone our acting skills, we can then much more easily apply these abilities to our real work as Jews.
So...break a leg, chas v'Shalom!