I have seen many articles that talk about a star's "habitable zone" as the ring around it which is just the right temperature for planets to have liquid water, which is necessary for life. My question is, why does a planet need to have liquid water in order to sustain life? Who says that extraterrestrial life can't live off of some other kind of sustenance?
Your question is not complete. I would suggest to slightly change the title to something like:
"Why must a planet be in a star's “habitable zone” to have life as we know it from our experience?"
You are right, when you are assuming that there could be another than carbon based life as we know it from our planet Earth.
There are many hypotheses that life could be based on Si or other elements.
But carbon is still the easiest element to create bounds with other elements and carbon too. Why? Here is the answer from wikipedia Carbon-based life:
There are not many other elements which even appear to be promising candidates for supporting life - for example, processes such as metabolism - but the most frequently suggested alternative is silicon.3 This is in the same group in the Periodic Table of elements and therefore also has four valence bonds. It also bonds to itself, but generally in the form of crystal lattices rather than long chains. Silicon compounds are generally stable but do not support the ability readily to re-combine in different permutations in a manner that would plausibly support lifelike processes.
There is a great documentary from a physicist Brian Cox published by BBC this year called Wonders Of Life which deals with the basic questions how carbon based creatures were "born" on this planet ;) + other interesting things about life.
Brian talks there why the first organisms were created probably in water and how it could be.
And why is water the best liquid to do the job? For one thing, it dissolves just about anything. "Water is probably the best solvent in the universe," says Jeffrey Bada, a planetary scientist at the Scripps Institution of Oceanography in La Jolla, Calif. "Everything is soluble in water to some degree." Even gold is somewhat soluble in seawater. (Before you get any ideas about extracting gold from the oceans, I should add that, according to Bada, the value of dissolved gold in a metric ton of seawater comes to about $0.0000004).
If you take a route that there could be a different form of life outside of habitable zone, then many things are possible. E.g. life on asteroids, ouside of solar systems, life in other dimensions, life based on silicon or some other elements, energy based life? Or even dark matter and dark energy based life or gods/God-like creatures and all kinds of sci-fi ideas ;).