There are lots of significant differences between the case interview and the fit portion interview, but one of the critical, key ones to consider is that the fit portion of an interview is driven by you. Yes, the interviewer may ask you about this or that job / hobby / accomplishment on your resume or they may ask you about a time you demonstrated leadership / collaboration / etc. But all they've done is ask the question, you get the latitude to pick the best story that will ultimately answer the question. Now, while it's true you have latitude, don't attempt to go all "smoke and mirrors" and tell a story which doesn't answer the question. We've all seen politicians dodge a question - don't do that! But do pick the very best story you've got at hand.
One great way to do this is to go through your resume (and career to date) and think about both the highlights and the lowlights. If you forced yourself to summarize the resume (which in turn should be a career highlights summary) without the use of school names and company brands, what key things would you highlight.
The key is to tease out the defining moments of those experiences. The removal of brand / school names here is critical, while it might be nice that you went to Harvard or worked at Google before business school, at most it's just a signal. For example, what lasting mark did you leave on Harvard or Google? What did you accomplish in your time there. If you're struggling to separate the wheat from the chaff, consider asking your friends or family or significant other (basically just someone who knows you well). The question should be: what stories about me stick out to you? This exercise has hidden benefits. If other people remember details, likely the story is not boring and has some "sticking power."
You don't need a fancy or outlandish or extreme story. What you want are genuine, authentic stories that highlight your accomplishments and talents. Some candidates fear that if they don't have some unbelievable story, they won't get due attention so they inadvertently stretch the truth. If you take a story and exaggerate or significantly stretch the meaning of it, it will become obvious to the interviewer that you're embellishing and likely backfire. Don't get caught in a trap where you're reading marketing copy on the firm's website and trying to fit a "square peg in a circular hole" with your personal anecdotes. Authentic stories do the best job of highlighting your real talents.
"Sometimes candidates come into these interviews and treat it like a video game - as if there is some sort of secret code and if they just hit the buttons in the right order, they'll get the offer." - Steve Kenning, former Bain & Co consultant
Generally, people have really good BS detectors so find your good stories and stick to them. Also, in those stories, don't be afraid to paint an authentic view of yourself. Everyone knows no candidate is perfect and what resonates is often authenticity. For more on the power of authenticity, see our detailed discussion with former Bain employee, Steve Kenning.
Finally, ensure the stories you are selecting have a great impact that you can point to. Specifically, you want to find examples where you significantly changed the outcome for the better. Would the same result occurred if you hadn't been involved? Did the event result in meaningful learning for you, your team, your company or your industry? If yes to any of those, you've likely found a great story to tell. It's important to remember that a story where you've got clear actions that drove a great impact is almost always better than a story about a phenomenal result you were affiliated with (eg my team/company/division accomplished some incredible goal). The consulting firms don't want to hear about success stories you were loosely affiliated with, they want to hear about success outcomes that you specifically drove.
For a deeper dive on the above topic, check out the below summary from the RocketBlocks Founder.
Telling vs. selling
Once you've got the right stories selected, you need to spend time thinking about not just how to tell the story but how to sell that story to the firm. You not only want them to comprehend the key pieces of any given story, but you want them to be thinking: "Yes, that's exactly the type of attitude / skillset / drive / leadership that we could've used on my last case!"