Political map of the gained provinces. Trøndelag in purple.
Initially, the Swedish delegates also demanded Akershus from Norway, which back then included most of southeastern and inland Norway, including Christiania/Oslo, but backed off that demand.
It is possible for armies to move between Tröndelag and Jämtland, and small armies fought over the province of Jämtland 1611-13, 1644-45, 1657 and 1677 while Sweden attempted to take Tröndelag from Jämtland 1718.
Taking Tröndelag in the peace secured the border and moved the Swedish strategic position forwards, ensuring the safety of Jämtlandn from Danish-Norwegian attacks. It was also believed that in the long run northern Norway would slide into a Swedish sphere of influence. The right to tax the Sami of northern Scandinavia had been hotly contested between Sweden, Denmark-Norway and Novgorod/Russia for a long time.
Sweden demanded Tröndelag for several strategic reasons:
It secured the border in the north, removing a route of potential invasion.
It is a relatively rich province that the Danes were more willing to part with than Akershus.
It carries to potential of Sweden gaining the sole right to tax the Same and northern Norway eventually becoming Swedish be integration.
Source: Den oövervinnlige (the undefeatable) by Peter Englund.