I've heard that castles must have towers, in particular at the corners where two walls meet, for defensive purposes. My question is in what way did towers benefit the defenders? What were the advantages to a tower over just having a wall?
Towers served as an elevated platform that provided a higher field of view and wider field of fire over the surrounding area. Being taller than the surrounding walls, it allowed defenders to rain down fire onto nearby walls to aid in their defense. Towers more often than not extended off from the walls as well, which provided for a wider field of fire than a flat section of wall may have provided. Additionally, a defensive wall may not be fully walkable along its length, so a tower provides a means for defenders to fire down upon attackers without having to have battlements along the entire length of the wall.
For a good example of this, take a look at the walls of Avila, Spain. The towers extend out from the walls at regular intervals and are slightly taller than the wall behind them. Extending out from the walls allows defenders in the towers to have a wide field of fire - in the case of Avila, about 180 degrees - to protect the lower and thus more easily scalable walls on either side. Towers are often placed close enough together so as to allow their fields of fire to overlap slightly. This allows towers to cover eachother's blind spots and for defenders on one tower to provide covering fire should an adjacent tower come under attack.
Though it may not seem like it at a glance, this kind of defensive philosophy would evolve into the Star Fort concept, with high, narrow towers morphing into low, broad bastions. Simple examples include Goryokaku Fortress in Hakodate, Japan, or Fort McHenry in Baltimore, Maryland. Though gunpowder artillery made traditional walls obsolete, the basic premise of the bastion remained as a highly refined version of a tower. They provide a vantage point from which to fire down on attackers, while their architecture - flat surfaces, sloping (but rarely vertical) walls, and defensive earthworks - is intended to minimize "dead zones" where defenders can't fire on while maximizing the amount of firepower a defender can bring to bear on any given point.
If I may, I’d like to suggest you take this question over to r/warcollege. They have lots of military historians that can help with this kind of question.