Because if you don't capture the castles, then you have a series of nests of enemies that can be a permanent thorn in your side. Yes, you control the countryside, but that just means your armies are wandering around, without any permanent base. Enemy castles can send out forces to harass your forces, then retreat back inside their walls. If you're invading, your aim is generally control of the territory. You can't have full control until enemy forces are stopped from harassing your occupying force.

It was difficult for an enemy force to maintain any kind of occupation without control of local fortifications, and this actually was one of the reasons that Wessex developed the Burh system (Not quite castles, but a similar concept) to allow them to both protect the population and permanently bother any large viking marauder bands. The vikings very rarely resorted to sieges, so the tactic was fairly effective.

To add to what James said, castles in particular, or fortifications in general, can also protect an important asset in their own right. They can deny an enemy the use of a strategic river, mountain pass, harbor, etc. The fortified town and castle of Conwy in Wales, for instance, was built to protect the mouth of the Conwy River, which helped give Edward I access to northern Wales.

If you're invading a country, you might need to take fortifications that block your advance in the mountains, or hold a port that you need to bring in supplies.

To add what has already been said in medieval Europe there had been the switch from national standing armies (as seen in Roman times) to mercenary armies made up of mostly foreigners. Mercenary armies are notorious for just leaving the army if they are not given adequate supplies (mostly food). This becomes an issue when campaigning far away from home as these resources become scarce. The only way to alleviate this was to let the troops forage the countryside for food. This didn't really work as when foraging troops would leave any ways. Campaigning was not very comfortable so troops would leave when given the chance.

Because of these changes in the style of European warfare armies became fairly immobile as large baggage trains were needed for any extended campaign to support the army without the need for foraging. Because they couldn't move fast enemy castles could disrupt supply lines if ignored and an army could not reach the enemy cities before this became a major issue.

As well, because of the low mobility of armies in this era defending armies had plenty of time to get into good defensive position and inhibit enemy movement. The only real way to stop this was to disrupt enemy supply lines and supply depots and to take these fortified positions.

Basically the way that European armies were designed at this time did not allow them to just simply bypass the enemy forts so instead armies would beseige them. This trend would continue in Europe until Napoleon who basically used an early version of the blitzkreig and rushed enemy capitals, forcing the enemy into unfavourable engagement. His army was the first European army to use troops with a sense of nationalism and gathered them from within France itself.

EDIT: If you want to read more on this subject both The Cambridge History of Warfare by Geoffrey Parker and Makers of Modern Strategy by Peter Paret have fairly good information on this subject.