I'm amazed that a three-year-old question hasn't addressed the most obvious answer to this topic:
Firebenders rely their environment, just like everyone else
Gabe's answer is partially correct, in that the internal state (emotions, discipline, etc) of a Firebender has a great impact on their ability to Firebend. However, we see that this is true with other benders as well: until Aang summoned a certain amount of stubbornness, he couldn't Earthbend, and until Korra learned to "be the leaf," she couldn't Airbend. Firebenders draw their "energy" from within, but only in a meditative, spiritual, and emotional sense.
No bender can create anything, even fire, from nothing. They draw it from their environment.
But without a source, they're powerless.
The power of Firebending, as in the actual energy that is erupting into flame, is dependent on the ambient heat/warmth sources around the bender in question. Firebenders consistently become weaker (or even lose their bending entirely) when cut off from sources of warmth and fire, and they are super-charged when near great sources of heat and energy.
We see this demonstrated again and again, throughout the franchise:
During the Siege of the North, it is established that Firebenders are stronger during the daytime, when they can draw power from the Sun. It is explicitly stated that Waterbenders, who are more influenced by the Moon, are more powerful at night.
During the Day of Black Sun, a global solar eclipse, Firebenders lose their bending entirely. When the single greatest source of planetary warmth is blocked, Firebenders are powerless.
Every 100 years, Firebenders become super-charged due to the presence of Sozin's Comet. The animations make it clear that Sozin's Comet is a HUGE ball of fire, big enough to light up the entire planet a deep red for hours at a time. It's no coincidence that the presence of a huge source of fire influences the benders of that element.
At the same moment, during the liberation of Ba Sing Se, Iroh says "Only once every hundred years can a firebender experience this kind of power." Note that language: he's experiencing the power, feeling it as an external source or event, it's not "generate this kind of power" or "create this kind of power."
When faced with the task of imprisoning a powerful Firebender, P'Li's jailers locked her in an ice-prison. Zuko specifically says "the extreme cold prevents her from Firebending," and P'Li herself notes "I've waited thirteen years to feel this warm" just before she bends for the first time.
That's the key: she'd already been freed and unchained, but she still couldn't bend until Zuko's dragon breathed fire at her. As soon as there was warmth nearby, she could bend again.
...it's just harder to see.
Think about fire, in real life. It's not a single thing. It can be different colors based on what's burning, radically different temperatures, or even completely invisible (such as Hydrogen fires). Embers can radiate plenty of heat to burn you, without any visible fire coming from it. A flame is simply what happens when the heat grows intense enough in one spot that matter and particulates in the air begin to glow.
Fire is a function of the concentration of heat.
Perhaps a more accurate name for Firebenders would be "warmth-benders" or "heat-benders", but ultimately they work just like the other three nations: they move and gather their element. In the case of Firebenders, they gather ambient heat, and usually produce the effect of erupting flames in the process.
But they're not generating flames, so much as they are bending heat, which tends to produce visible flames as a side-effect.
In fact, Waterbenders can work the same way.
Generally, when we see benders work, they are moving visible quantities of an element from one place to another. But there is another example of someone "producing" their element, seemingly out of nothing: Hama the Bloodbender demonstrates the ability to pull water from any living thing, gathering a substantial (and deadly) amount of water from a field of flowers.
This is the perfect illustration of Firebending: just as Hama isn't "creating" water, she's simply gathering small amounts together from across a wide area, so Firebenders can't "create" fire, they simply gather the ambient heat around them and concentrate it.
But without even those minor sources of water, Hama is powerless, just like P'Li was powerless in her ice prison.
One Final Note
This idea, of Firebenders "breaking the rules" by creating fire instead of bending it, was one of the things that M Night Shyamalan changed in his ill-fated movie adaptation The Last Airbender.
In that film, Firebenders are generally only shown bending fire from something, like a nearby campfire. When Iroh demonstrates the ability to actually create fire, seemingly from nothing, the nearby soldiers react with amazement.
But, as in so many other ways, M Night had misunderstood his source material: Firebenders don't create heat, they bend it, and in the process they make fire.
