Why have sieges in warfare from ancient times through the medieval era and later consistently lasted months or even years, and what sudden events or circumstances typically allowed attackers to finally break through?

Majority of famous wars and battles throughout history feature a siege, where a city or fortress is surrounded by enemy forces.

From ancient conflicts such as in Troy all the way to modern times with Xiangyang, they seem to consistently take months to years for the the forces to finally break through enter.

Why would they take this long? What factors made it so they suddenly get inside?