If we take the official novelisation as gospel, the strict answer is that Maul's initial successes (in holding off the Jedi) were because of his martial prowess, his training regime and his personal hatred of the Jedi, versus an older opponent and a relatively untrained opponent. His ultimate downfall was his overconfidence.
Apologies for the great big lumps of text.
- At the start of the fight the two Jedi seem quite evenly matched to the single Sith:
But on this day, he [Qui-Gon] had met his match. The Sith Lord he battled with Obi-Wan was more than his equal in weapons training, and he had the advantage of being younger and stronger. Qui-Gon was nearing sixty; his youth was behind him and his strength was beginning to diminish. His edge now, to the extent that he had one, came from his long experience and intuitive grasp of how an adversary might employ a lightsaber against him.
Obi-Wan brought youth and stamina to the combat, but he had fought in only a few contests and was not battle hardened. Together, they were able to hold their own against the Sith Lord, but their efforts at attack, at assuming the offensive against this dangerous adversary, were woefully inadequate.
Darth Maul was a warrior in his prime, never to be any better, his powers at their apex. In addition, he was driven by his messianic hatred for and disdain of the Jedi Knights, the enemies of the Sith for millennia. He had worked and trained all his life for this moment, for a chance to meet a Jedi Knight in combat. It was an added bonus that he was able to engage two. He had no fear for himself, no doubt that he would win. He was focused in a way that Qui-Gon recognized at once—a Jedi’s focus, mindful of the present, locked in on what was needed in the here and now. Qui-Gon saw it in his mad eyes and in the set of his red and black tattooed features. The Sith Lord was a living example of what the Jedi Master was always telling Obi-Wan about how best to hear the will of the Force.
- As the fight progresses, both sides are largely trying to find a chink in their opponents fighting style while taking as few risks as possible:
His agility and dexterity allowed him to keep them both at bay, constantly attacking while at the same time effectively blunting their counterattacks, relentlessly searching for an opening in their defense.
Qui-Gon pressed hard in the beginning, sensing how dangerous this man was, wanting to put an end to the combat quickly. Long hair flying out behind him, he attacked with ferocity and determination. Obi-Wan came with him, following his lead. They had fought together before, and they knew each other’s moves. Qui-Gon had trained Obi-Wan, and while the younger Jedi was not yet his equal, he believed that one day Obi-Wan would be better than he had ever been.
So they challenged the Sith Lord quickly, and just as quickly discovered that their best efforts were not good enough to achieve an early resolution. They settled into a pattern then, working as a team against their enemy, waiting for an opening. But the Sith Lord was too smart to give them one, and so the battle had gone on.
- After briefly tussling, Maul takes a heavy fall to the floor that leaves him "visibly stunned".
Down the Sith Lord tumbled, landing hard on a catwalk several levels
below Obi-Wan. The force of the fall or perhaps the unexpectedness of
it left him visibly stunned, and Qui-Gon leapt down after him, sensing
a chance to put an end to things. But the Sith Lord struggled back to
his feet quickly and raced away, taking the battle in a new direction.
- Qui-Gon loses primarily because his opponent is younger and faster:
Stroke for stroke, Qui-Gon and Darth Maul battled about the rim of the
melting pit, locked in a combat that seemed endless and forever and
could be won by neither. Then the Sith Lord parried a downstroke,
whirled swiftly to the right, and with his back to the Jedi Master,
made a blind, reverse lunge. Too late, Qui-Gon recognized the danger.
The blade of the Sith Lord’s lightsaber caught him directly in the
midsection, its brilliant length burning through clothing and flesh
- And Obi-Wan secures victory because he attained a new level of connection with the Force. Maul also failed to capitalise on his tactical advantage and was clearly overconfident when fighting an opponent he considered to be essentially disarmed.
Eyes fixed on the Sith Lord, Obi-Wan Kenobi went deep inside himself,
connecting with the Force he had worked so hard to understand. Calming
himself, stilling the trembling of his heart, and banishing his anger
and fear, he called upon the last of his reserves. With clarity of
purpose and strength of heart, he launched himself away from the side
of the pit and catapulted back toward its lip. Imbued with the power
of the Force, he cleared the rim easily, somersaulting behind the Sith
Lord in a single smooth, powerful motion. Even as he landed, he was
drawing Qui-Gon Jinn’s fallen lightsaber to his outstretched hand.
Darth Maul whirled to confront him, shock and rage twisting his red
and black face. But before he could act to save himself, Qui-Gon’s
lightsaber slashed through his chest, burning him with killing fire.
The stricken Sith Lord howled in pain and disbelief.