The most popular scripts in Europe are Roman, Cyrillic and Greek. I find it surprising that a continent with such a huge linguistic diversity would use only a few scripts. This is striking in comparison to the Indian sub-continent which has a diverse set of Abugida scripts. Were there any scripts which died out during the course of history?
Well firstly, Europe doesn't have huge linguistic diversity at all. It's easily the continent with the least linguistic diversity.
However, that's kind of irrelevant because there is no direct relationship between type of script and language. What script people use to write down a language is entirely dependent on choice. Therefore, the predominance of the Latin and Cyrillic alphabets in Europe is, as CommodoreCoCo points out, historical happenstance.
This is partially because of the pervasiveness of the Roman Empire. India never really had anything like it. The Gupta Empire, for instance, controlled ~3.5 million km^2 for ~250. Rome ruled for ~450 over 6.5 million km^2 at its peak. The largest empire, the Maurya, controlled ~5 mil km^2 for only ~140 years.
Yes, some scripts have died out. However, the writing systems Latin replaced were largely Runic alphabets. These are conceptually identical to the Latin alphabet and ever share many recognizable letters. This made a transition very easy.
Actually, bear in mind that the various Abugidas have all evolved from one single script known as the Brahmi script (http://en.wikipedia.org/wiki/Brahmi_script), and hence all belong to the same family of scripts. It's a bit like both the Roman and Cyrillic scripts being derived from Greek, which in turn was derived from Phoenician, and so hence belong to the same family of alphabets.