When people say structured, they often mean they want to see you being MECE - be able to look at a problem but comprehensively, but also in clearly defined categories - I think the "CE" is more importance than the "ME" - it's tough to be perfectly mutually exclusive sometimes.
With initial frameworks, they typically involve some combination or permutation of products/competitors/customers/market (highly generalized, but you get the idea). With profitability cases which come up often, you typically see something like external/internal frameworks with external being all the market/customer/competitors factors and internal being the Profit = R - C, with deeper exaimination into company specific factors. The above are highly generalized, but the idea is to give the sense of a "here is a categorized way to look at all the major dimensions of the problem", and then you proceed with the analysis. Case in Point and other popular books have good ones and memorizing them and practicing customizing them to the situation enough times will make you solid enough.
Now, WITHIN the case is trickier because we aren't always as conscious of it, but it can be thought of in a similar way. Here's a simple strategy I would try - whenever you are asked a question, try to categorize it in a logical way and use that categorization to think through your answer.
Example: How can we increase sales?
Unstructured response: "We can do A, B, C"
Structrured response: "Since sales are price x quantity, let's look at ways to optimize pricing, drive quantity, and then maybe even expand our product portfolio. Within pricing we should consider.... To drive quantity in our existing products we should.... Finally we can expand our portfolio to...."
Example 2 - What are some ways to improve our company's image in light of the recent negative publicity?
Unstrucrued - We can advertise, we can donate money...
More structured - We can either focus on improving the image and impact of our products, build our company's outreach activities, or highlight our own people strategies. By products, we can advertise them as representing strong values, highlight the environmental or ethnical ways in which they were produced, or the positive impact that our products have on customers. For our company's outreach, we can become more civically involved through engaging in communities efforts, CSR programs, participate/host conferences on key issues. Finally, we can highlight the positive programs we have for our own employees such as paid family leave, flexible work policies - in fact, many tech firms have generated very strong PR by highlighting how they treat their employees.
Note - I'm not saying the above are the best structures - I'm sure better ones are out there. But by creating the structure, it makes the problem/situation feel more manageable and guides the conversation better.