This combines two separate phases in the history of string theory.
The anomaly cancellation defines the rules for how the strings interact in 10 dimensions.
The Calabi-Yaus come later. The strings are still moving in 10 dimensions, but instead of having 9 infinitely large space dimensions, 6 space dimensions are closed up in a subatomic donut shape. So there are only 3 large dimensions for the strings to move in, but on the very smallest scales, the 10-dimensional rules are still at work.
A Calabi-Yau is a type of 6-dimensional subatomic donut shape which is "flat" in a way that makes it stable. Specifying the Calabi-Yau defines a background space for the strings to move in.
Originally they didn't know if Calabi-Yaus are mathematically possible. But they found a few, and eventually thousands. Still, there was a hope that the real world might be based on a special Calabi-Yau.
Meanwhile, the astronomers discovered dark energy. Around 2000, Bousso and Polchinski suggested a way to get dark energy from string theory. There are fields coming from the strings, called "fluxes". Bousso and Polchinski said you could have flux wrapped around the donut shape in different ways. Most ways you could do that, would produce dark energy so strong that galaxies (and thus life) couldn't exist. But if you did it just right, there would be a big cancellation of positive and negative energy, with only a little excess left over, as the observable dark energy.
Steven Weinberg had suggested, even before dark energy was observed, that in a large universe or a multiverse, you could have the amount of vacuum energy varying from place to place. But observers could only find themselves in a place where the vacuum energy was weak enough to allow galaxies to form. Bousso and Polchinski found a way this might work in string theory - different flux configurations in the Calabi-Yau. Then a few years later, various collaborations ("KKLT", "KKLMMT") started producing concrete scenarios.
If your 6-dimensional donut shape has hundreds of possible paths for the flux, and if about a dozen of them are used, then the number of possibilities is about 10^500. That number somehow became the standard talking-point for the size of the "landscape" of possibilities.
So to recap, we have one string theory, an unknown number of Calabi-Yau possibilities, and then an absolutely enormous number of flux possibilities on each Calabi-Yau. And then we have the proposal that cosmic dark energy comes about, because the vacuum energies from the fluxes in our Calabi-Yau, cancel just enough to make the existence of galaxies possible, and they had to cancel for any observers to be here - anthropic principle.
But now the worry is, what about all the other parameters of observable physics, like the masses of the particles? Can you also tune them by playing with the details, the way that the vacuum energy might be tuned? This is where the idea that string theory can "predict anything" comes from: no matter what we see, somewhere in the landscape, there will be a Calabi-Yau and a set of fluxes which produces that physics.
I don't know how true that is and I don't think anyone does. I think the usual expectation is that, even if our Calabi-Yau has hundreds of "handles", that all the standard model particles are strings that are wrapped on just one or two neighboring handles, whereas the vacuum energy comes from everything. So tuning particle masses means working with details localized in one corner of the Calabi-Yau, and you might not have the same flexibility, as when you are adding up positive and negative contributions from all the fluxes.
Meanwhile, if you are just trying to get particle physics from a Calabi-Yau, and you're just ignoring dark energy for now, this isn't how you work! You don't choose a big Calabi-Yau, then populate it with junk fluxes, and then try to make them balance. Instead, you pick something simple and symmetrical, and try to make the results look roughly like the standard model, which is already hard enough.
So the string landscape looks something like this. There is this enormous, intimidating, and possibly infinite range of possible geometries, most of them incredibly complex. People who like the anthropic principle say that we live in a universe or multiverse where the Calabi-Yau varies from place to place, and we are in a corner where the fluxes were tuned to make life possible... But most string research that aspires to experimental relevance (string phenomenology) focuses on parts of the landscape where the 6-dimensional Calabi-Yau geometry is simple. These are the people who are just trying to find something that matches experiment. And it might be that even the right value of dark energy can come from something simple - that it's not anthropic at all.
One of the ultimate questions is, what cosmology is implied by string theory? The anthropic people like "eternal inflation" because that is a theory in which there are infinitely many localized big-bangs, and each one of them could have a different Calabi-Yau and a different set of fluxes, so it's a cosmology in which anthropic tuning of physics could occur: all those other big bangs happen, but logically we had to be in a big bang which made our existence possible.
But we don't actually know yet, whether this is the right way to do cosmology in string theory, according to the theory's internal logic. Inflation is mostly a field-theory concept, and field theory is just an approximation to string theory. In my opinion, the big question in quantum gravity right now is how to do quantum gravity in de Sitter space (accelerating expanding universe), and theorists are still fighting over that one. So maybe we live in a multi-big-bang universe; but maybe we don't, maybe there was just one. The real optimists would hope that there was just one big bang and it led inevitably to just one point in the landscape. Then you would have truly unique predictions.
A lesser possibility is that there was just one big bang, but it can lead to many places in the landscape, depending on the details of the big bang. So string theory overall wouldn't make unique predictions, but there would be a set of separate string cosmologies, each of which individually makes unique predictions.
Finally you have the anthropic multi-big-bang scenario. This way of thinking has become synonymous with string theory and the landscape, in some people's minds, and it might be what the theory predicts - when you finish working it out! - but that isn't proven yet.
It might seem to be taking a long time to sort all of this out, but in the history of physics, a few decades is a reasonable period of time, to work out the details of a really fundamental advance. If you look at the 20th-century history of quantum field theory, sometimes a decade would pass and people were just stuck on something really basic; and in fact we're still discovering important mathematical properties of QFT. The string saga will eventually reach its natural endpoint, when we finally have the basics of the subject sorted out. For the present we can say that string physics is capable of qualitatively imitating the real world - producing particles and forces like those we see - and that it might be capable of quantitatively matching the real world, too. But we still don't know how unique or inevitable such an exact match would be, because we don't understand string cosmology.
