In one episode of Star Trek: TNG "The Traveler" appears and the Enterprise achieves speeds over Warp 10 (as I recall). The Traveler has kind of superpowers to master space and time (all pretty weird in this episode :)) and becomes a friend of Wesley. At the end he tells Picard to further Wesley, as he has special abilities and is an abnormal boy. Is this explained more in detail in other episodes and where do these abilities come from?
The Traveler described Wesley Crusher as:
a genius with the intricacies of time, energy and propulsion and that those talents needed to be encouraged.
It was also stated:
he had a unique intelligence and great potential if provided encouragement and opportunity, comparing him to a child prodigy like Mozart.
I imagine it was his superior intellect that enabled him to go beyond the normal human limitations and achieve power over space and time.
The Traveler returns in two episodes after his original 1st season appearance - once, to help Wesley rescue Dr Crusher from a warp bubble he had created by accident (Remember Me - season 4), and the second time to apparently lead Wesley away and teach him the knowledge of the Traveler's species (Journey's End - season 7).
From these later episodes, it would seem that Wesley had an innate ability to understand the connections between time, space, and thought, and also had an almost intuitive feel for warp field geometry.
Whether he's "the next step" or simply a very talented/gifted individual isn't really made clear.
In Season 7 Episode "Journey's End" it is revealed that Wesley's abilities include the ability to stop time, or travel outside normal space-time existence.
The question also asks where the abilities come from. While the traveller indicated that Wesley had evolved to a new level of existence, certain episodes suggest certain factors that may have led to the conditions that allowed this evolution to take place.
In Season 4 episode "Remember Me" Wesley is experimenting with a warp bubble and accidentally traps his mother in it. However, his mother does not realize that this has happened, and unknowingly re-creates her entire universe out to a distance of a few hundred meters, taking most of the remainder of the episode to realize what she had done. While Wesley created the warp bubble, I think it is important to note that Doctor Crusher physically created a reproduction of the universe using only her mind, thereby indicating that she too had special abilities that Wesley had inherited from her.
Furthermore, one might ask where these special abilities in Doctor Crusher came from. In the Season 7 episode "Sub-Rosa", we learn that a long line of Doctor Crusher's maternal ancestors (20 generations!), terminating with her grandmother, were "possessed" by an anaphasic life form named Ronin which used the Howard family candle to survive. It is highly probable that the merging of this anaphasic life form with so many successive generations of hosts introduced mutations in the DNA which Wesley inherited from his mother.
Wesley is essentially hinted to be a next step in human evolution. He is purported to have the ability to develop powers similar to "the Traveler" and does end up leaving with him when Wil Wheaton left the show.
As to what those powers actually are, or why they were given to Weslet, it isn't ever stated.
This is hinting at the start of Wesley becoming the first Q. The angelic Traveler is sent to wake up his abilities which lead Wesley to eventually realizing his full potential - all-powerful, omnipotence.
After much time Q-Wesley becomes lonely and thinks back to his most cherished memories, as a mortal on the Enterprise. Disguising himself, he finds that he can relive some of those moments - popping in on his old friends from time to time...