It's all right there in the episode.
As Anwar stated, the Enterprise-C did not cause the war (I believe Castillo or Garrett even said that there was a peace treaty in the works when they left). In the normal timeline, it responded to the Klingons' distress call and was destroyed while fighting off the Romulan attackers of Narendra III. When the ship was destroyed, two things happened:
1. Survivors of the Ent-C were captured by the Romulans (including the alternate Tasha Yar).
2. The Klingons saw the Ent-C's sacrifice on their behalf as honorable, and decided to become friends with the Federation.
When the Ent-C was taken out of the equation because of the spacial anomaly, two things happened:
1. There were no Ent-C survivors, because there was no Ent-C to battle the Romulans.
2. Because there was no Ent-C to battle the Romulans, Narendra III was completely destroyed. Even if the Klingons there had known that the Ent-C at least tried to answer their distress call, they all got killed when the Romulans attacked.
Because in the altered timeline there were no witnesses to the attack, it became a non-event, and at some point later on relations between the Federation and the Klingons deteriorated. Whatever peace treaty had started pre-2344 went down the toilet, and the two powers started a war.
When alternate-Picard sent the Ent-C back to 2344 (with alternate-Yar, who was always supposed to go back too), the timeline corrected itself.
It's just like the "Data's head buried in 1800's San Francisco" thing. In JJ Abrams's nuTrek, Data's head is still buried in San Fran in the JJverse (because it shares a common past with the Prime unverse) even though Data will most likely not even exist in this new timeline (or his circumstances will be radically different than the prime timeline).