The phrase "Eating Bitter" is a term commonly used in China, particularly in practing Gung Fu or martial arts. Its definition lies in the belief that through hard work and pain, success and strength are the result. There is a Chinese proverb which states, "You cannot know sweetness until you have eaten bitterness."
Learning martial arts, especially in a traditional sense, is not easy. The body and the mind are often pushed to limits which are quite uncomfortable. The teacher may break you down to rebuild you into something new, and your body will be bruised, bleeding and sprained. Through this we may know discipline and end up befriending it, learning from it, and using it in our daily life. Not unlike what the military does to us.
In Gung Fu and Taiji, the forms are meant to be "moving meditation," entering what is known by Buddhists as Samadhi. Samadhi is becoming one with the practice, focusing the entire mind on one thing. This is not easy, but that's why it is a practice. The martial arts, I understand, are not for everybody, but there are many lessons from the practice that can be applied to life.
We as people "eat bitter" often in our daily lives. Our jobs may become intensely difficult, our home life may become stressful, and so forth. We learn from suffering, and value the sweetness of the lesson. Suffering brings enlightenment.