I was watching a nice little documentary on Castles on the BBC the other day, and the presenter was talking about castles being a force multiplier, and saying things like "Whoever controlled Stirling Castle controlled the whole of Scotland", or "Edward's castles in Wales exerted English control over the country".
It occurred to me that I'm not entirely sure exactly how that worked - how the existence of a castle in Stirling can have an effect on the surrounding countryside (even allowing for hyperbole like "the whole of Scotland"). I'm assuming that soldiers from the castle can patrol the area, block roads and so on, but it's always just sort of expressed as a given...
Roughly yes, castles were usually placed near major towns and transport routes (harbour, river crossings or passes) allowing the garrison to control the local population and watch for enemy troops while having a safe location to wait in. More importantly they could control the flow of people/trade though the local trade routes and enforce taxation allowing you to influence more than the surrounding area (if stuff can't get from A to B because of C then C can influence both A and B)
In the case of Sterling its the control of the last downstream crossing over the river Forth which basically cuts Scotland in two. If you can't control Sterling you couldn't really use the crossing and would have to make a major detour into the hills to find another crossing. If you were an rebel Scottish army of any great size such a detour would give the English time to raise a counter army and march to meet you.
Well I'm by no means an expert, but I've recently been studying the use of castles in Wales in particular so I might be able to add some insight for that specific example.
First off the castles that Edward captured, improved or built in the years of the Welsh Conquest (roughly 1277-1290s) had a larger purpose than just holding the territory. Rhuddlan castle, for example, was intended from the first as a military and administrative center for northern Wales (Gwynedd). As such it was constructed so as to be able to house the administrative machinery of Edward's household and performed this purpose well, especially in the 1282-3 campaign. In fact, the nearby river Clwyd was dredged and redirected to allow naval navigation up to the walls, so that it could be continually resupplied by sea in the event of a siege. Beaumaris and Harlech castles are similar examples of this idea. This allowed Edward to keep forces garrisoned throughout North Wales regardless of an uprising, as the Welsh were usually ill-equipped for a full-blown siege action. In this function they were used as much for military bases as they were for territorial dominance.
Caernarfon castle is a good example of an administrative center, and was used to not only dominate the countryside but also to house the shire-government after the reorganization of Gwynedd into shires based on the English system.
What's more, the building of many of the castles in Wales coincided with the establishment of townships throughout the area. English colonists were given special rights in exchange for colonizing these towns. In some instances the towns were little more than villages attached to the castles, but in the case of Conwy specifically extensive town walls were attached to the existing castle so that the town could be better protected. Thus castles were used to protect the new urban centers, which quickly became centers for commerce. By protecting and dominating these towns the English were able to continue their dominance of the local countryside. One should consider that many laws were passed after the 1282 war that basically restricted all trade to these specific towns, so the Welsh were required to bring all their goods to them in order to sell. Essentially the castles exerted influence far beyond their walls in this way, as they protected the primary centers of Welsh commerce.
Some historians have suggested that the sheer magnificence of the Edwardian castles in Wales may have cowed the native Welsh into submission, as such buildings were not very common in Medieval Wales prior to the conquest. This seems to not really be the case however, as two rebellions followed the establishment of English control over Gwynedd despite the presence of several castles by that time, and that was just during Edward's rule. Wales continued to occasionally burst into open rebellion, such as during the rising of Owain Glendwr.
Michael Prestwich, who wrote the primary biography of Edward I, suggests that the overall effectiveness of the castle-building campaign is questionable, especially when one takes into consideration the immense cost of building such massive fortresses. Some, like Rhuddlan, did provide a mustering place and center for military organization during rebellions, but castles like Harlech and Criccieth were of dubious importance and cost large amounts of money to keep supplied once they were beleaguered.
If you have any questions please let me know and if I can help I'd be glad to!
Hope this was somewhat helpful. Some good resources on the subjects are:
Prestwich, Michael. Edward I. (London, 1988).
Morris, J.E. The Welsh Wars of Edward I. (Oxford, 1901).
Davies. R. R. The Age of Conquest: Wales 1063-1415. (Oxford, 1987)
I asked a slightly different but closely related question a little while back, which has some useful information in the responses.