The brain teasers on this site are more like math problems or logic puzzles. You're not likely to encounter those in an interview.
More common in interviews (though not as much in case interviews) are brain teasers like, "how many golf balls can you fit into a passenger Boeing 747?" This question requires you to break down the problem area (the airplane's volume, which is made up of the cockpit, cargo hold, passenger area, etc.), make assumptions about it (the cockpit contains X cubic meters of space, the passenger area Y units, etc.) and do calculations (you can fit Z golf balls in X cubic meters, etc.). These questions test your ability to define a problem, break it down, and use math to solve it.
In case interviews, you more often see a business-oriented version of these, often called "estimation" or "back-of-the-envelope" questions. Though they often require similar techniques, estimation questions tend to focus on a market (e.g., the number of women's jeans sold per year or the vodka revenue in one night at a high-end night club) and are used to estimate the impact of a consultant's recommendation, often within a larger case.
For example, you may be completing a case with a clothing retailer and find that volume sales have declined significantly within a certain region where customers have turned to online shopping. You recommend the company invest in online retail platforms or form a partnership with an online retailer. Your interviewer sees your reasoning and asks you what the impact to the business's financials will be. You then have to estimate the revenue generated and compare that to the expected investment costs. Doing so allows you to take a recommendation ("partner with an online retailer") and actually calculate how much it helps you reach your client's goals.
As you prepare for your interview (and consulting itself), you can keep your skills sharp using the brain teasers on this site, but you'll want to really master estimation questions as they apply to business problem-solving and case interviews.