How exactly does slighting a castle work?

To quote Richard Nevell, an archaelogist who did his thesis on the subject:

“One of the key challenges in identifying slighting is that the methods used to slight a castle were the same used while attacking it. For example, mining or undermining was used in siege warfare to bring down outer walls. But you could use the same approach to demolish a castle. This can be seen at Bungay Castle (Suffolk) where excavations in the 1930s identified an unfinished mine gallery underneath the great tower. Documentary sources show that the King had ordered its destruction, but it was evidently reprieved at a late stage.”

Given that the objective is to render the structure incapable of being used as a fortification, the focus would be on those aspects of a castle that give it defensive advantages: 

• the outer walls, which allow a small garrison to hold off the enemy from a safe distance. 
• the towers, which provide further height and protection compared to relatively open ramparts.
• gatehouses, which protect the most important entryways with portcullises, arrow-loops, machiolations, and murder holes.

Depending on how emphatic the slighter wanted to be, they could leave the rest of the structure intact and allow it to continue to exist as a

château

(a non-fortified stately home), or they could render the interior unlivable and let the ruins be subsumed back into nature. 

Generally, slighting would be accomplished by mining at the base of the structure in question and then burning down the props of the tunnel (i.e, undermining). Later on, the process was sped up enormously with the use of gunpowder, which could be straighforwardly detonated in mere minutes. For the destruction of interior elements, pulling down roofs and then firing the interior was usually sufficient.