The foundational skills to demonstrate are:
The Ability to Draw Conclusions from Data
The Ability to Synthesize or Communicate Your Conclusions
It's important to describe how you arrive at your conclusions.
I have both done case interviews myself as well as conducted them. The key for being prepared is checking out sample case interviews online and practicing as much as you can. There are certain structures for how to conduct case interviews in the most effective way, so check those out as well. It's always important to take your time, ensure you are communicating clearly to the interviewer what your thought process is, and making sure you ask any clarifying questions if needed.
In terms of practicing, choose some sample case studies, grab a friend, and run through the case with them. It's important to practice cases out loud as this can make a big difference in how you practice your communication and articulate your thoughts. Lastly, I would take into account what type of consulting you are interviewing for. For example, if you are interviewing for a technical consulting position, you may have a case study that also has questions on how to introduce or implement a new technology and what your strategy for that would be. If you are looking at management consulting, it might be more financial and numbers focused where you might have to do some quick math during the case study.
At the end of the day, practice is very important and remembering that a case study is just a problem you are being asked to solve- which you do everyday!
When it comes to case interviews, I would recommend doing three things:
1. Familiarize yourself with the "rules of the game"
2. Read up on common business thinking
3. Practice, practice, practice
Maria has drawn a really good outline on what the "rules of the game" are. When doing case interviews, understand that the interviewers are trying to assess your ability to break down a ambiguous problem (such as "Company X is experiencing a decline in profits. Why?") and structure it based on individual components of the business. I would say once you get the Problem Structuring part of the case interview mastered, the rest should be relatively easy to follow-through.
When it comes to structuring the problems, it would definitely help to read up on common business theory. This includes things such as the profitability framework (profit = revenue - costs). Additionally, definitely read up on the different components that a business will consider when developing its strategy. This commonly includes the 3Cs: cost, customers, and competition. To aid in your development of business theory, I highly recommend investing in resources that will directly teach you the materials. Maria has given a really good website that teaches you the fundamentals of case interviews. I would also recommend Case in Point and CraftingCases. The latter you can Google and watch their modules for free!
Lastly is practice. Case interviewing is a skill on its own, and through much practice you will get better at it. I also recommend practicing with people who understand the game too; you wouldn't want to learn basketball from a football coach!
My final tip is don't get discouraged! When I first practiced case interviewing, I felt so incompetent because I can't answer a simple market-sizing question. Nevertheless, keep pushing through and sooner or later you'll understand.