When referring to a literary experience, one is most likely meaning reader-response. Reader-response is based upon an individual's interpretation of the text, its impact on them as a reader, and the text's ability to engage the reader as an active participant in the text.
Therefore, in regards to the literary experience "we" could gain from Kipling will, most likely, never be universal. Not all readers will experience They in the same way.
For me, given your question poses a truly subjective answer, my experience with They reminds me that I am a human being. What I mean by this is that human beings are subject to pain, happiness, life, and death. The fact that Kipling retells a true story of his life in They, one can easily reflect upon how he or she cannot control death. Kipling lost a daughter and he continues to see her, and other deceased children, throughout the story. This forces me to reflect upon the life of my own child, how I do not relish her life enough on a day-to-day basis, and that I could see myself having a similar experience trying to keep her in my world.
Kipling's love for his daughter does not allow him to let her go. His love for her extends far past that of the physical world.
In the end, my own literary experience is one which forces me to reflect upon my life and remember to not take any minute for granted.