As the fire (and other signs) progress through the wheel of the zodiac, they take the original energy of that element and go on to develop increasing complexity in how it is expressed.
It is thought to be an expression over time (as expressed spatially across the wheel of the zodiac). It is also the result of combining the signs element (fire) with its modality (cardinal, fixed, mutable).
So for the fire signs this would mean that Aries (a cardinal sign) could be considered the most “pure” expression of fire. It is outward reaching (as fire rises), it considered passionate, willful, quick to anger (as fire is hot), etc. Also, Aries is known for being concerned with the self and having a very strong ego.
The same strong ego drives and outward expression are present in Leo (a fixed sign), only at this point in the zodiac they have become a bit more tempered or sophisticated. Rather than being concerned with will, aggression, and dominance, Leo tends to show its fiery side with more of a focus on self-expression, creativity, and play.
Ego elements are still there, of course: Leo’s are known for being very proud. But the focus is more on creativity and self-expression rather than ego dominance.
By the time we get to Sagittarius (a mutable sign), that fiery energy has been tempered even more. This sign still possesses the fire element qualities of outward directed expansion, which is why it is so heavily associated with ideas like travel, adventure, and exploration. This also includes the themes associated with exploring new horizons in ways that are mental or spiritual (for example, through religion, philosophy, etc.).