By the time a soldier reaches the rank of sergeant and earn their own nickname, you will mostly have developed some level of attachment towards him/her after watching the raw recruit developed into a true badass hero. Therefore it makes it absolutely heartbreaking to watch them killed in battle, especially if it is the result of poor decision, since you are fully aware of the fact that YOU are partly responsible for the death.
It is worst if you are the type of role-player that imagine backstories and personalities for your soldiers.
For even more fun, watch them joke, exercise, and visit their injured comrades in between missions. It's a small addition to the anthill interface, granted, but you grow really attached to them after seeing the soldiers get along.
When a Council Member leaves the Project. From their point of view anyway; They sank all their hope, all their trust, everything they had into this project, actively working with hated enemies and political adversaries to keep it funded, all to save Humanity from the brink of extinction. Yet it wasn't enough, the guns fall silent, the populace they're responsible for are killed in the unreported terror attacks or taken by the Abductors. Everything they put into the project failed to save them. Their trust waivers as their first, and last, line of defense has abandoned them. Soon they must abandon their post to see to their own defense, throwing in all the strength they have left. Perhaps they regret their choice to support XCOM and lament that under these bleakest days, humanity cannot avoid it's fate.
Some players take to renaming medals after fallen troopers they had gained an attachment to. One such medal is earned when a soldier dies (and a second time after losing five soldiers!) and is often renamed in this manner.
Some missions, Terror Missions in particular, can place you and your soldiers before some very Sadistic Choices. Do you move in to save a set of vulnerable civilians and put your troops at serious risk, or play it safe and watch the victims get slaughtered?
An even worse version can come up with explosive weapons. A heavy soldier might have a cluster of chryssalids right in rocket range - but with no way to hit them without also killing the civilian they have surrounded. One can only imagine how traumatic an experience like that might be for the heavy.