How did the alliance system drag many countries into the war

While the importance of alliances cannot be understated they were not the primary cause of The Great War. It was the unconditional support of Germany for Austria-Hungary for whatever course of action it would decide concerning the assassination of Archduke Franz Ferdinand. Austria-Hungary would have never went to war without Germany's support.

It is also important to note that the decision for Germany's unconditional support laid solely on the shoulders of Kaiser Wilhelm II of Germany. While he tried to change course of his actions when he realized the magnitude of what was about to start it was just too late.

The link below presents in an illustrated manner that is easy to follow the large context of the war.