I experience for the American officers and soldiers that friendship which arises from having shared with them for a length of time dangers, sufferings, and both good and evil fortune.

This quote is saying that the speaker has built a connection of trust and friendship with the American officers and soldiers that he/she has spent time with. This camaraderie was built over a period of time because of experiences that the speaker and the officers and soldiers had together, such as facing and dealing with dangers, suffering and the good and bad luck they encountered.