First of all, is there any truth to this claim? Second, what made him so good?
Yes. Although he came close on several occasions - e.g. being almost killed by Polish rebel, then near Ochakov by charging Janissary (heavily wounded), then in Swiss Alps
He really wanted to be a great general - since he was 5. Oddly, his poor health rather helped. He was quick to catch cold in winter, with small statute, physically weak. So his father did not sign him up to serve in Imperial Guards regiment in childhood (as it was a custom among prominent noble families - while child was growing, we was promoted). Instead, father was trying to push Alexander into religious career. This push wasn't completely wasted - Alexander was quite religious Orthodox Christian and even made some improvements in Orthodox service, always liked to sing in church choir, observed all fasts etc. It also played well for rapport with soldiers - who were en masse more religious than officers.
But Alexander persevered in his desire to become an officer and by luck a close family friend convinced Suvorov's father to register Alexander for military service.
He trained hard to strengthen his physique, learned many languages (as a rule, he was able to speak to natives in all campaigns).
Because of late start, Suvorov experienced soldier's life for a couple of years. It gave him deep understanding of soldier's needs. He also was a voracious reader and thinker. Those qualities were amazingly rare among russian officers at a time.
Sources
(russian)Петрушевский. Генералисcимус князь Суворов: в 3 томах = A. Petrushevsky, "Generalissimo Count Suvorov" (St. Petersburg, 1884), v. 1-3. Still most definite source about Suvorov's life.