For the record, Races of Eberron was my introduction to the Eberron campaign world. I knew little about the world before reading RoE, and it got me interested in the setting as a whole.
While there is a lot of Eberron-specific details, you could easily use them as stepping stones to integrate the races into a non-Eberron world. While the culture and descriptions mention Eberron-specific places and religions, you could probably replace those with similar places or gods from your own campaign. If you don't have anything that matches then you may have to re-write some of the history, but you have to expect to be able to do that if you're taking things from one setting and putting them into another.
I hear a lot of people say that the warforged are hard to put into an existing non-Eberron world, but honestly I don't see how hard it could be. They could be the ancient armies of an extinct race, just now unearthed. A modern wizard could invent them. They could come from one of the planes. I could come up with a lot of ideas on how to put them in.
The kalashtar would be the hardest to put into a non-Eberron world, IMO. The idea behind them is that they're basically the result of a merge between human souls and quori, which are outsiders from the realm of dreams (which is not a pretty place at the moment). To add them without significantly changing their background you'd have to find a type of outsider that uses psionics, give them an excuse to run away from their homeland, and make up a reason why their fellow outsiders can't easily chase them. The kalshtar lose a lot without the quori, Dal Quor, the Inspired, and the Dreaming Dark. To fully integrate them into another campaign you'd have to import all of those ideas as well, or find something to replace them.
I do have to say, though, that the only thing I thought was missing from the book was good physical descriptions of some of the races. Specifically, I remember wondering what the shifters looked like. Sure, there were pictures, but I like to have written physical descriptions of what the average member of a race looks like and how that might vary from individual to individual. RoE seemed to assume you had already read the campaign setting as far as looks go, though they did include full statistics for each of the new races.