Being fire signs, Aries, Leo, and Sagittarius need something to fuel them and their drive. They’re dependent on it, just as actual fire is.
Aries’ fuel is simple, yet powerful. They’re fueled by the future, by everything that lies ahead of them, and ultimately by themselves. Aries’ actions are necessary to start the journey into the zodiac, they’re “brand new” which makes it their duty to start the race. The cycle begins with them, making them only have themselves and the future. While this may seem as if that would make them never run out of fuel, that’s not true. They and the future are inseparably connected, and if they should stop trusting and believing in one of those, the other would be dragged down as well. Being the starter, the igniting spark, they have no other choice than to put trust in their moves going further. If they wouldn’t, their fire would soon burn out.
Leo’s fuel is to no surprise validation. Acknowledgement. This may seem simple, but it’s not - Leos need to be seen and shown a reaction. They’re a fire sign, yet fixed, they’re ruled by the sun which represents the glowing and confident individual, but they’re in the shadow of their sister sign Aquarius, and with that in both Uranus’ and Saturn’s as well. Every form of acknowledgement they take in like a sponge. Leos are known to be drama queens, and that may be true, but they need to be. It gets them the validation they need. It is in no way a positive one, but that doesn’t matter - they won’t be ignored, they have to be reacted to, and that’s what they need. They’re bound by their shadows and their fixed nature to not be able to run all alone, so their flame depends on the acknowledgement of those around them.
Sagittarius’ fuel is their search and, above all, possibilities. They complete the fire cycle, being mutable and having the planet of expansion floating above them - they start the last round through all the elements of the zodiac. Sagittarians don’t have an end-goal. Their destination is their journey and their arrow always points to what they, ultimately, can’t reach. Nothing is ever finished for them, there’s always something more, something not yet found. That’s what makes subjects like religion and philosophy “theirs”, they can’t find definite answers, there’s always another possibility, another unexplored idea. Sagittarius’ fire burn out when they settle - not physically, but when their mind stops wandering, because that’s their purpose. They started the last big journey of the zodiac, the way into the higher realms. They can’t settle for that would then put an end to their flame.