The CLT states that as we draw random samples from a population, the distribution of their means tends towards a normal distribution.
Not quite, though maybe you have a book that says something vague and not particularly accurate like this. Such vagueness of language has misled you about what it is even referring to.
It's not the number of samples, but the number of observations in a sample. Sample sizes are very large in typical A/B tests, but see the later discussion, which explains why that might not be sufficient for the sort of variables commonly used in many A/B tests. First lets look at what the CLT says, or at least let us get a bit closer to a formal statement of it.
In particular (for a 'classical' CLT in mean-form), if $\bar{X}_n$ (n=1,2,3...) is a sequence of sample means of $n$ independent and identically distributed observations from a population with finite mean and variance ($\mu$ and $\sigma^2$), and $Z_n = \frac{\bar{X}_n-\mu}{\sigma/\sqrt{n}}$ is the standardized mean, then in the limit as $n\to\infty$ the (cumulative) distribution function, $F_n(z)$ of $Z_n$ converges to the standard normal cdf, $\Phi(z)$.
(Conveniently, this theorem - relating as it does to distribution functions - is in a form that is potentially relevant to evaluating tail probabilities.)
This would suggest as sample sizes become very large, the distribution function $F_n$ of a statistic $Z_n$ should eventually become close to that of a standard normal distribution. The CLT itself doesn't tell you how large that might need to be; it only talks about what happens in the limit as $n$ goes to infinity. In some situations (even when the CLT holds), that might need to be very large indeed.
In particular, you might consider very skewed distributions (which are very common in calculations like click-through-rates or purchase rates or whatever, and also in effectively continuous quantities such as time or money spent on a site) and see that sample means can sometimes remain clearly non-normal even when sample sizes are getting into the thousands.
Nonetheless, the difference between these two samples is guaranteed to be a normal distribution.
I presume you intend "difference between sample means" there (otherwise, where does the CLT come in? 'difference in samples' is not a test statistic until we define how to do that), but either way, this is wrong. Indeed if the original distributions were non-normal you can guarantee that the distributions of the sample means (and of their difference, given the usual assumptions) is not actually normal. However, it might in practice get quite close $-$ close enough that the normal will yield perfectly reasonable answers, except perhaps in the extreme tail $-$ if only the sample size were large enough. Large enough, that is, given the particular situation you're in and your particular sense of what might be close enough.
My question is, how is the difference between these two samples constructed, and why is it guaranteed to be a normal distribution?
Look to the specific statistic you're using in the test. You don't mention which it is, and in A/B testing there's at least two distinct situations that are commonly involved and many more which might be possible. For both those common cases the statistic at least has a numerator with the form of a difference of means.
However, the CLT alone is not sufficient for the whole statistic in either case. A suitable argument for a t-test (e.g. if you were testing say, time or money spent at a site) or a z-test (as an approximation of a binomial test of proportions) would require additional argument, since the denominator is also a random variable. Such an argument is possible (e.g. by invoking Slutsky's theorem).
