how does dominion expand your empire?

[Taqwus]

Makes it easier, mostly. Dominion makes a +/-1 modifier for morale, for instance, and affects numerous pretender and prophet stats, plus some spells have greater effect in one's own dominion.
In addition, some of the dominion scales are such that you only benefit from the positive scales if they're from *your* dominion, and you might be happier in your dominion than somebody else's. For instance, regarding magic/drain, Ulm smiths ignore drain scale for research purposes. If it pushes drain into somebody else's lands, 'tho, that might hurt their research. Heat/cold can also be pushed usefully... sometimes very valuably, e.g. Caelum ice armors perform much better in Cold +3 than Heat +3. And Ashen Empire Ermor dominion is literally quickly lethal...