Short version: Numbers aren't everything and it is entirely possible Yuan Shao entered the way at a disadvantage despite the traditional narrative. That it was a camapign of a warlord seeing things turning against him and seeing to strike before it was too late but was beaten by a better commander in a well prepared position

Traditionally, it goes Cao Cao was the superior leader, calm headed, wise, able to keep camp united, listening to wise advice, deceive, and so so handsome (ok maybe not that last bit) vs Yuan Shao, weak, indecisive, ignoring the best advice and listening to bad council, arrogant. Cao Cao may have had so so few men, Yuan Shao's army of supposedly 100,000 infantry and 10,000 cavalry in size but the abilities of the two men won out.

Yuan Shao failed to strike due to the illness of a son when Liu Bei had rebelled in Xu then during the campaign, Yuan Shao barged ahead against the advice of Ju Shou and Tian Feng, fighting an aggressive but mostly one-fronted campaign and committing himself across the Yellow River. He chose the brave captain Yan Liang for a role above his abilities, he had the misfortune of soon losing Wen Chou, demoralizing his army in the first two skirmishes. He jailed Tian Feng, he demoted Ju Shou, he ignored Xu You's bold plan to strike at Xuchang who then defected with vital information to Cao Cao. When Cao Cao struck at Wuchou, where the supplies were based under Chunyu Qiong, he ignored Zhang He and backed Guo Tu's advice to concentrate forces on Cao Cao's base at Guandu. Cao Hong and Xun You held out, Chunyu Qiong was overcome then executed and without supplies, Yuan Shao's army collapsed with Zhang He defecting, Ju Shou unwillingly captured.

Guandu was Cao Cao's iconic victory and Wei historians built it up for all they could. Yuan Shao became the representative of the old guard, the old nobility, arrogant, too lenient on the power, relying on family fame. Cao Cao representing the change, the dynamic representation of the new age, the figure who overcame terrifying odds via his superior leadership and character. Wei distanced Yuan Shao's role in Cao Cao's rise (they would also claim victory in a battle Cao Cao lost to the Yuan sons), they (at the very least) exaggerated the number gap between the two sides when the conflict came and they built a narrative around it.

The message of the two leaders can be summed up in the ten points by either Guo Jia or Xun Yu (gone with his version) as Cao Cao considered if to give after Lu Bu, whether they said or not it shows the message that would be portrayed for the campaign itself (translation Yang Zhengyuan)

Considering the successes and failures of ancient times, those who truly had ability,though they were weak they were certain to become strong, but those who did not, though they were strong they became weak. Liú [Bāng] and Xiàng [Yǔ] in their successes and defeats are enough to demonstrate this. Now the only one who can contest the realm with you is Yuán Shào. [Yuán] Shào appears outwardly lenient but inwardly is jealous, appoints people but suspects their intentions. You are of wise judgment and adaptable, and only ability determines appropriate [appointment]. This is superiority in judgment. [Yuán] Shào is hesitant and makes few decisions, missing opportunities. You are able to decide great affairs, and adapt to changes without limit. This is superiority in planning. [Yuán] Shào’s military discipline is lax, his orders are not firm, and though the soldiers are many, infact they are difficult to use. Your orders are clear, rewards and punishments are enforced, and though the soldiers are few, they all fight to the death. This is superiority in military. [Yuán] Shào relies on his family accumulations, feigns appearances of wisdom in order to collect famous followers, and therefore many gentlemen of little ability but good reputation follow him. You use benevolence to care for people, sponsor sincerity and not empty good appearance, in your own conduct are solemn and frugal, but in rewarding the accomplished begrudge nothing, and therefore the truly loyal and upright gentlemen of the realm all wished to serve. This is superiority in virtue. With these four superiorities to support Heaven’s Son, wielding justice to go on campaign, who dares be disobedient? What can [Yuán] Shào’s powerful words do?"

Now as for the campaign itself

Part 1