The answer to this varies depending on the obstacle that we are talking about.
Some obstacles were dealt with through technology. The Europeans, for example, built ships that allowed them to sail across any ocean in the world. They also created weapons that allowed them to subdue enemies that they ecountered.
Other obstacles were dealt with through sheer ruthlessness or even evil actions. For example, Europeans lacked a work force for their plantations in the Caribbean. Without a work force, their expansion would have been useless. They dealt with this problem by enslaving Africans.
Finally, some obstacles were dealt with through clever use of the natives of the places where the Europeans expanded. For example, Europeans might use one nation of Native Americans to help them subdue another nation that stood in their way.
All these methods were used at one time or another to deal with obstacles that stood in the way of expansion.