As Dave mentioned, a very similar question on this topic was asked about a year ago, which is worth looking over. I happen to be one of the respondents, and you can read my answer here.
The following is going to be in rough terms aimed at making things easily understandable. For more details and technicalities, see my answer (and attached comments) that I linked above.
The CLT essentially states that the distribution of the mean of the data will become normal as the sample size increases, it says nothing about the distribution of the data itself. So you could have normal, exponential, bimodal, ect. distributed data but, if your sample size is large enough, the distribution of the mean will be normal.
For example, say you are measuring a metric from a population that is exponentially distributed. Imagine you take 100 random samples from the population where each sample is large enough for the CLT to apply. If you were then to take the mean of those 100 samples, the CLT essentially states that those 100 means would be normally distributed. So the repeated measures of the mean are normally distributed, despite the fact the underlying data is actually exponentially distributed.
If this is still a bit confusing, I personally found this (quirky) video very helpful as it does a great job illustrating these concepts.
So why does this all matter?
Knowing the means are normally distributed allows us to use a variety of parametric tests (anova, t-test, ect.) which operate on the assumption that this is true. This is both the power and pitfall of the CLT: the ability to compare non-normally distributed samples to one another.
I think where this gets confusing is that just because you can compare two sample means to each other doesn't mean that you should. (ie comparing the means of two exponential distributions might not tell you what you think it does, or two bi-modal distributions, or a bi-modal with a uni-modal distribution, ect).
The question most people should ask is, "is the mean (or a difference in means) a useful metric given the distribution of my data". Only if the answer to this question is yes, should one proceed to compare means (thus relying on the CLT). This is where those Q-Q plots and tests for normality you mentioned come in. They examine the distribution of the data itself to see if it is normal. This is important as the distribution of the data is critical to keep in mind when interpreting the results of a statistical test. If the data does not follow the distribution you expected it to (in this case normal) that comparison of means likely doesn't tell you what you think it does.