The "other world" here can be taken somewhat literally as the new geographic surroundings around the bridge which Robert Jordan surveys at the beginning of the book – they already have a certain fatefulness about them, since they will be the site of his risky mission. It can also be taken as the new group of hardened, diverse guerillas he meets, with which he has to build relations and trust quickly. Or, on the most abstract (and "deepest") level, the new world is the world of attachment to others, which he's never really felt before. This is in both the friendships he begins to develop with Anselmo and Pilar (and to a lesser extent the rest), and the instant and powerful attraction he feels for Maria. The world of the guerillas is upset, too: Pilar supports Robert Jordan, and replaces Pablo as the leader of the group. Finally, there's a tint of the supernatural, which Jordan doesn't take seriously, when Pilar reads his palm and sees something ominous (threatening).
Starting that first night, and continuing through the second day, Robert Jordan experiences things he never has before, and is caught up in them. When Maria comes to sleep with Robert Jordan on the first night, it dramatically opens up a new dimension of life for him: love. He's surprised by it, and doesn't know yet how deep it is; he also has trouble believing it really happened throughout much of the next day. The next day, on the way to El Sordo's, Robert Jordan also begins to form a tighter bond with Pilar, whose story about the killing of the fascists in her hometown horrifies and captivates him and builds his admiration for her as a story teller and a comrade. In the afternoon, he has revelatory sex with Maria, in which both of them feel the "earth move." This too barely seems real, and Pilar interprets it in mysterious, somewhat supernatural terms.
When Robert Jordan returns from El Sordo's, he finds a somewhat jealous and irritable Pilar, an apparently drunken and obnoxious Pablo, and a snowstorm. The snowstorm spells frustration: if it doesn't stop, it's hard to see how they can manage their mission, and if it does, El Sordo's sunk. Later on, a confrontation with Pablo ensues, and it looks as if there might be gunplay. It does stop snowing, so El Sordo is sunk. Bad news. Although Robert Jordan still has the positive experience of his love with Maria to cling to, his overall attitude grows progressively more worried at this point.
As Robert Jordan predicted, the next day El Sordo's band is bombed, and none of them survive. With the numbers of his people cut in half, it begins to seem to Robert Jordan as if they will all die in the mission, though they might still accomplish it. It also becomes clear that the fascists have anticipated the larger offensive, and will be prepared; Robert Jordan tries to warn Golz, but without much hope. Then, when things couldn't possibly get much worse, they do: while they're sleeping on the third night, Pablo steals the detonating equipment and escapes. Robert Jordan feels desperate, and starts to resign himself to a mission that's all but certain to fail and cost them all their lives. Plus, on their last night, Maria won't sleep with him, because she's not feeling well.
At the moment of total despair, Pablo saves the day by coming back with reinforcements and restoring the group's confidence. They are able to complete the mission, and most of them are able to escape. Though the end is certainly thrilling, we can't call it a successful escape and return, at least not obviously: Anselmo is killed, the Republican attack is a failure (because Andrés couldn't stop it in time), and Robert Jordan loses the new love of his life…and his life. But in another sense, Robert Jordan does escape successfully. As he faces death, he realizes how much he has learned, and recognizes the way in which the bonds he formed gave him a new reason for living. And for dying. Though he dies, his transformation has been a successful one, and Maria and his friends, who are in some way supposed to be him, go on.