The issue of the origin Slavic ethnic identity is currently still hotly debated. The crux of the issue is what you regard as 'slavic': are you talking about a group of material culture (particularly a specific type of dress-pin, a bow fibula), or about a specific language (the Slavic language group), or about how people might have called themselves (Huns, Avars, Slavs)? Depending on which of these criteria you choose to focus on, you can see three different 'origins' of the Slavs.

The historical mention of the name of the Slavs is easiest. We can find that in the late 5th century at the Byzantine Danube border, near modern Hungary, where Roman writers are referring to groups of Sclavenes. However, it seems that these Romans are referring here to what they perceive as a group that places themselves as distinct from Huns, Avars, and Antes. Because this 'group-making' is contextual (it seems that in the breakup of the Roman and Hunnic empires, people were looking for new identities), it also makes sense that in the next centuries, new groups formed out of them (such as the Bulgars Bulgarians, or the Moravians). This is simply a matter of politics, and has little to do with what anthropologists and modern laymen would consider 'culture'.

If you are looking at language, however, a completely different picture emerges. Just as we can do for many related languages, we can reconstruct old Slavic languages and identify a region where this language must have originated (based on local loan-words, names for vegetation, and linguistic diversity). In this, there are two basic camps: those who identify an area in modern Southeast Poland, and those who identify a region in modern Western Ukraine. As an archaeologist, I am only dimly aware of the issues in slavic linguistics, but we do have other people around who can provide more details on that.

The third issue, of material culture, is also still contested by archaeologists. The primary 'marker' of Slavicness is the bow fibula, but they date mostly from the later period of Slavic ethnogenesis, the 6th and 7th centuries. It is also possible that these are not Slavic at all, but Avar, as their greatest concentration was found in Northern Ukraine. Recently, this approach of naming a specific artefact as a 'marker' of ethnic identity has come under severe criticism. Does drinking Coca Cola make us Americans? Not necessarily, and similarly, within 'Slavic' areas in Eastern Europe there is in the 6th century quite a diverse range of different fibula types.

So in conclusion, the process of the spread of Slavic cultures is much more complicated than simply a single group 'spreading out' over all of Eastern Europe. There were a variety of peoples who at different times and for different reasons adopted Slavic customs, including the language, and who by historic times identified themselves (and were identified by others) as 'Slavs'. This does not mean that some areas were not settled by Slavic peoples through migration and conquest, but simply that this was not the only way in which a large part of Europe ended up Slavic.