They are like children because war has destroyed the foundation upon which they understood their lives. It's like starting over again, learning to perceive and understand the world again with a new set of parameters. They are like old men because of what they have seen, it has taken their innocence.

They are lost because they don't know what to believe in now. If their previous life has been destroyed, what is left for them? They feel that they cannot return to the life they lived before, but don't know what is next, assuming they survive war.

This quote is one of the reasons why this book is such a classic war novel. In this one quote alone, the author manages to capture the whole gamut and range of emotions, which are quite paradoxical, that are experienced by soldiers fighting on the front during WWI. Note the way that the forlornness of the soldiers is juxtaposed with the experience. They are at one and the same time both like children and old men.

The second answer is indeed excellent, and there isn't much to add to it. I love the way the passage works with opposites and paradoxes:

“We are forlorn like children, and experienced like old men, we are crude and sorrowful and superficial—I believe we are lost”

Somehow, simultaneously, they are like children and like old men.  They are crude but they are also sorrowful (which implies deeper emotion than crudeness does), but they are somehow also not only sorrowful but also "superificial," which returns us to the kind of shallowness already suggested by "crude." Finally, notice the nice ambiguity of "I believe": Paul doesn't say that they are lost, he merely says that he thinks they are lost. The whole passage is brimming with splendid ambiguity.

Even with the life experience of old men, it is still possible to find oneself "lost" in the world, without a major purpose or reason to continue. A young man who has the knowledge or mental age of an old man -- through trauma or a hard life -- may find his efforts "superficial" because he does not believe them to have meaning.

(Personal: I find this quote profoundly depressing.)

After the experiences on the front, Paul feels that the "tender, secret influence that passed from them [their desires and scenes of their youth] into us could not rise again." For, with the war and its killing fields, they have become detached from their other lives:

But it would be like gazing at the photograph of a dead comrade; those are his features, it is his face, and the days we spent together take on a mournful life in the memory; but the man himself it is not.

There is no longer any "comradeship with the things and events" of their existence that ihas been cut, making things incomprehensible to the young men.  They are lost in the events of the present, united only in the "course of [their] days."  Truly, part of Paul and the other young men has died.

This detachment that Remarque writes about is not unlike the detachment and disllusionment of Hemingway's characters who have survived World War II.

This quote describes the physical youth of the soldier ("children"), but the age of the soldier's psyche ("like old men"). As with your other quote, there is the reference to that which is lost when a person takes the life of another in "war." "Forlorn like children" indicates confusion on the soldiers' part. Though society has given its approval for the killing under the cover of war, it makes no sense to the young soldiers. As they try to come to terms with the taking of life (and the threat of death), their experiences change them:

...we are crude and sorrowful and superficial...

They see nothing civilized in their actions and they feel lost because they cannot go back to a time when they saw the world through the happy haze of innocence.

The previous answer is very nicely done, quite thorough and detailed.  I will add to it only the idea that this quote is pretty much summing up the idea of the entire book.  One of the major points of the book is that war destroys the people who are exposed to it.  Even those who live are forever changed.  They are fundamentally lost no matter what happens to their bodies.

Here, again, Paul is trying to capture the sheer number of conflicting emotions that plague him day and night.  If you break it down into its components, you can make sense of the whole:

"Forlorn like children," = pitiful, sad, abandoned, lonely, and very much without a sense of direction (either in the war or in life).

"experienced like old men," = these soldiers experienced more death in one day than most experience in a lifetime.  Add to this the kind of raw instinctual wisdom it must have taken to survive, and in every way, though young, these boys feel very old.

"crude and sorrowful and superficial" = certainly this one is more ambiguous, but contextually, consider that each resting day together is a gift.  They laugh and joke and play games, their humor is likely coarse or boyish "toilet humor" and because it is one of the only emotional means of escape (and infrequent), it is often crude.  But then, it is laced with sorrow because it is the reminder that normal still exists, somewhere, for someone.  The thought of another battle reminds them that this moment of normalcy is short lived, always.  Further, the death of many who may have been joking only days before is another moment of sorrow, as well as the realization that this might be the last time to joke.  Finally, superficial, because, in many ways, the light hearted times, wedged between battles at the front (and more death), are only passing moments that bring them back to death again.

"lost" = given the mixture of extremes, Paul's final assessment of the depth of his experience and emotion can only be summed up in this word.