Your insight is well-placed. The grandmother is not characterizing the granddaughter as a person of great maturity, extensive experience, or worldly inclination. What impresses the grandmother is how the granddaughter's intelligence and education (which she mentioned explicitly) have made her aware of how the contemporary world works. People of the grandmother's generation typically find it difficult to comprehend what is going on in the world or how to interpret the cascade of events because they lack the epistemological foundation. They feel challenged by how the contemporary world moves and turns and they are impressed by how easily the younger generation navigate the world around them with confidence. The younger generation's exposure to the media gives them a high level of literacy to interpret what is going on.

وذكية ومتعلمة، شايفة دنيا وعالم
". . . smart, educated, and well-versed in world affairs . . ."

Also: "exposed to world affairs" and similar expressions.