What made "The Great War'' a worldwide phenomenon?
The single largest contributing factor to the 'world-wide' aspect of World War I was entangling alliances. An alliance is when two or more countries agree to help each other out if needed.
Before WWI, many of the countries in Europe had formed alliances with each other. Some of the alliances were well-known, and some were secret. When Archduke Franz Ferdinand of Austria-Hungary was assasinated by the group the Black Hand, (Serbian Nationalists), Austria-Hungary declared war on Serbia. Austria-Hungary, a huge powerful country in Europe, felt pretty good about this, because Serbia is relatively small and weak. Austria-Hungary had already formed a powerful alliance with Germany, but they did not count on Serbia having an alliance with Russia.
So...this is what happened, because of the entangling alliances:
Austria-Hungary declares war on Germany
Russia announces their intent to moblize the army to defend Serbia
Germany views Russia's mobilization of the army as an act of war against Austria-Hungary and declares war on Russia
France (bound by a treaty to help Russia) declares war on Germany and also Austria-Hungary
Great Britain (in treaty that states a 'moral-obligation' to help France) declared war on Germany. Britain also had a treaty to help Belgium too, and Germany had invaded Belgium as well.
These entangling alliances drew in all the major countries of Europe. When you also factor in that many of these countries have multiple colonial holdings across the world, the war escalates from being a continental war to World War I.
Consider the colonial holdings/connections:
Great Britain: India, Canada, Australia, New Zealand, Egypt
Canadian and Australian young men and women signed up to the join the war effort on behalf of Great Britain. Beyond man-power, the colonies contributed supplies to the war effort and generally supported the mother country.
Ultimately, when the U.S. joins the war, the Great War has esnared both hemispheres, tying up multiple continents' attention and focus: North America, Europe, Africa, Australia, and even Asia.
The main thing that made the war a "worldwide phenomenon" was the fact that the European countries had colonies all over the world. This led both to fighting in many non-European areas and to people from all over the world participating in the fighting in Europe.
As examples of fighting outside of Europe, there was fighting over German colonies in the Pacific and in Africa. Because the Europeans had colonies (and, in the case of Britain, related countries like Australia and New Zealand), people from around the world were involved in the fighting in Europe. Troops from Australia and New Zealand, for example, were heavily involved in the terrible battle at Gallipoli. Support units (people not fighting but doing labor for the armies) from African colonies participated in the war in Europe.
In these ways, the Great War involved most of the world even though the bulk of the fighting happened in Europe.