In my experience, there are a few key things that students misunderstand around the central limit theorem.
Firstly, many students don't actually have a concept of distribution in the first place. They need the idea that anything you could record about an object/person/group has various possibilities for what the outcome could be, some of which might be more likely than others. The description of what the possible outcomes are and how likely they all are is called the distribution. This description is sometimes a table of probabilities, sometimes a formula, sometimes a graph; and sometimes, the pattern of probabilities is so common and so useful it has a name, and just the name is enough to tell a statistician everything they need to know. The normal distribution is one of these.
Secondly, students don't have any distributions in their heads that aren't normal! When they think of a measurement with a mean, the exercises they've done so far have required them to draw a normal distribution anyway. So it's hardly surprising or useful to them that the mean of the means is normal because they think, well isn't everything anyway? Therefore, it's probably useful to show some distributions of things that aren't normal so that they appreciate what it might mean for it to be normal.
Thirdly, and most importantly, many students don't have a concept that the sample mean has a distribution at all! To them the sample mean is one value they calculate so how could it have a distribution? It is important to stress that anything that can be recorded or calculated about an object or group of objects has a distribution. That is, there is a description of all the outcomes that could possibly be and how likely they are. This point is almost never made explicitly about groups of objects, and I feel it's a big gap in most stats teaching.
As to how to deal with these things, well, here is what I would do. I have only actually tried some of these things as presentations rather than as class activities, so if you do try them I'd love to hear how they worked out!
First, I would list several things you could record/calculate about a group of objects and show graphs of the distributions of these things: the mean height of a group of 6 people, the standard deviation of the IQ scores of 10 people, the median of the prices of 100 houses, the maximum of the potassium level in 5 bananas, etc. The point you are trying to make is that any measurement you could calculate has a distribution, and this distribution might or might not be something recognisable.
If at all possible, I would get everyone to do their own little sample. It could be as mundane as getting everyone to roll a die 5 times and record the result, or as interesting as getting them to measure the length of 5 snake lollies. If you can't get them to do it themselves, you could always provide everyone with their own 5 measurements, but I think it would help if it was an everyday thing they could measure themselves if they wanted to.
Then get everyone to calculate various things from their sample and record the distributions of them on a big graph on the board. I'd recommend doing a couple of crazy calculations, like "the smallest, plus the biggest, minus two of the middle one", just to really reinfoce that any calculation has a distribution. This should give them a real feel for how different calculations produce different distributions. It should be clear that the mean has a particularly nice distribution.
Then you could do it with a bigger sample size, maybe this time with just the mean. After having done this twice, it's probably ok to proceed by just showing them the results that you would have gotten rather than getting them to do it.
Now you can show how the distribution of the possible means is nicely clumped in the centre and more so as the sample size increases. And you can say that it's not just clumped in the centre, there is a specific well-known distribution that describes this shape -- the normal distribution.
A final comment: the fact that the distribution of sample means is approximately normal is actually not the point, of course. The point is that it's really useful to know that it will come out this way because then you can use everything that mathematicians know about normal distributions to say something useful. Doing all the preceding is probably a waste of time without some discussion of why you care so much about it!