I can't reproduce the exact numbers produced by this calculator but the output seems quite close to the result you might get from
power.prop.test() for a power of 50% in a two-sided test, using the sample estimates for the two proportions, e.g.
power.prop.test(p1=26/190, p2=25/200, power=.5, sig.level=.1)
(The result is 4390 per group, so 8780 total or 8390 visitors more than you already have, the 15-observation difference could easily be accounted for by a rounding error somewhere – including on the website – or some other computational detail.)
If this is really what it is, it seems that there is a fundamental flaw in the approach, as it implicitly assumes that the proportions you got from your sample are indeed the true proportions, which seems to defeat the whole point of doing an experiment in the first place. In fact, there is no telling if the difference will “reach” significance, as you could very well discover that the results are very different after you have collected more data (perhaps the difference will be smaller or even go in the other direction).
Intuitively, this should be easy to realize if you consider that the recommended sample size is more than 20 times the size of the sample you already have. The data you have will be dwarfed by these new data and you will in effect have a completely new experiment that could go either way (especially since it's already clear that the effect is not very strong and the data you already have are not conclusive – otherwise, you could already reject the null hypothesis).
For a similar idea, expressed more carefully in a completely different context, you might refer to Kraemer H.C., Mintz J., Noda A., Tinklenberg J., & Yesavage J.A. (2006). Caution regarding the use of pilot studies to guide power calculations for study proposals. Archives of General Psychiatry, 63 (5), 484-489.
The information provided on the calculator's page is in any case very thin and insufficient to serve as a basis for a proper power analysis. Even if you were willing to disregard the point I just discussed, you would at the very least need the desired level of power (together with the effect size and error level) to compute a sample size and it does not seem to be mentioned anywhere.