He seems to be fine once they get back to sickbay. The situation, however, is not: the probe starts moving towards them, they're too close to use photon torpedoes, they can't outrun it, and phasers don't seem to have any effect on it. (It's also sending out some kind of energy field which is in all probability threatening.) The day ends up being saved by Barclay, who channels warp power into shields in a previously unknown way, and strengthens the shields enough for the ship to be able to fire photon torpedoes safely and destroy the probe.
But Barclay's intuition, intelligence and confidence don't stop there. A short time later, he proposes reprogramming the Argus central computer virtually singlehandedly in two days, rather than fixing each reactor individually (a task of at least 2-3 weeks' length). He gives a virtuoso acting performance, wowing both Beverly and Deanna, and later makes a pass at Deanna in 10-Forward. Finally, Geordi finds him arguing grand unification theories with Albert Einstein in the holodeck (and holding his own, at the very least). This is enough to set him worrying, and he takes Barclay to sickbay, where Beverly finds that his brainpower has increased incredibly, making him "the most advanced human being who has ever lived."
Since Barclay's hardly done anything that could be considered menacing, Picard decides to let him do his work. This only changes when Barclay decides the normal computer interface is too slow to let him stabilize the array properly (which is true, as the reactors are about to all go critical), and hastily constructs in the holodeck a device which allows him to directly patch into the computer. In effect, he becomes the Enterprise computer-and by the time it's clear what has happened, his mind has expanded enough that forcing him back into his own body would be fatal.
Geordi, after hurried consultations with the bridge crew, gets to work on rigging a bypass that would at least let them move the ship to a starbase. Barclay, however, decides to use his newfound knowledge of speed and distance to manipulate subspace, creating a never before seen disturbance. He ignores Deanna's pleas to stop, and blocks Geordi's attempts just in time. He then manages to repel the attempt by Worf and a security team to forcibly remove him, and sends the Enterprise hurtling smack into the center of the disturbance he's created.
After a major shake-up, the Enterprise emerges right by the center of the Galaxy. The face of an alien appears, babbling nonsense, but a reconstituted Barclay explains: their race, the Cytherians, also explore the Galaxy, but they do it by bringing other civilizations to them, rather than traveling themselves. In effect, they "reprogrammed" Barclay in such a manner as to let him bring the Enterprise here-but they're benevolent, and only want to exchange information for a while. Several days later, the Enterprise returns to its own space intact, and Barclay settles down to being "plain old Barclay again," with Deanna's and Geordi's help.