Interesting question, and I'm sure it will mean different things to different people.
For my own training, it is much as Jun has described; my personal degree of self-reflection and self-challenge.
For example, I actively seek out uncomfortable situations (not just in my training) whether they be due to fear, pain, annoyance, dislike, frustration or anything at all that makes me feel uncomfortable. I seek these situations out to understand my reactions, learn why they occur, learn to be comfortable with them, and if possible or sensible, to transform them into something different and perhaps better as perceived by myself.
I consider the entire experience spiritual because it ties back to the entire human experience and I've found that working on my fears has been the single greatest influence on my training, my psyche, and my behavior and interaction with other people.