Ed makes the remark about "being alone" with the letter because there is a very good chance that it will bring bad news. And in fact Jethro and his mother are alone when he reads Tom's letter to her, which has the news that "Eb and me is alright." Even though they have survived the action at Fort Henry, Tom describes the brutality of the fight, and how one of his friends is burned to death. Tom's description of the hardship at Donelson—of how many of the battle-weary soldiers threw away their heavy blankets to lighten their loads, then froze to death when the weather turned—is perhaps even more chilling. His mother's response—her face "strangely twisted" with sorrow—is to wordlessly go into the pantry and close the door, to be alone with her grief. Even though Tom and Eb are alive, Tom's letter has made the hardship they face vividly real.
Tom's letter lets the family know that he and Eb are both still alive and well. They were both at Fort Henry, where the Navy handled most of the fighting, and at Fort Donelson, where casualties fell most heavily on the foot soldiers like them. Tom reports that he and Eb saw many of their friends killed or else badly injured. Tom's letter also suggests that his excitement at the prospect of joining the battle has faded in light of the things he's seen and done:
I am not so proud about Donelson as mebby I ought to be...you tell Jeth that bein a soljer aint so much.
Ed Turner is right that it's best for a family to be alone when a letter from a soldier arrives; the news it contains is never certain to be good (what if Tom was writing to say that Eb was hurt or dead?). As it is, even with both his brother and his cousin alive and well, the letter still fills Jethro with "a great loneliness."