Firstly, people here should understand that e-voting as in voting machines and internet voting are completely different and not really comparable.
One of the opposition parties of Estonia is strongly against internet voting, mainly because their voters are not using it a lot and they are able to mobilize their voters well to go voting on paper as opposed to most other parties. For various reasons they are in power at the capital city and the trip of the researchers to go and observe the current voting process was paid by the city, so already for that they can't claim that they are totally independent. And, of course, the fact that the whole thing came to light a few days before the elections of the European Parliament was just a coincidence. This far they have yet to actually publish the report, which, from what we know this far, doesn't have any new attack vectors, only the ones that were already considered more-or-less from the very beginning.
Estonia has a smardcard-based ID card that can be used for authentication and digital signatures (two different keys). The latter is legally as good as your handwritten one which means you can build all sorts of services on top of that, elections are just one of them. The vote is encrypted with the public key of the current election, signed with the ID card and sent to a central server. Later, the double votes are removed according to the list of people who voted on the election day (so if you were forced to vote for someone and your ID card taken away, you can just grab your passport and go vote again using the paper-based method), votes are separated from the signed container, moved to a physically different machine, decrypted and counted. Anyone can go and see how all the process is done, it is fully auditable and all the video recordings of the whole process are later uploaded to Youtube. By no means it is so that only some certain people are chosen to make the audit to get favourable results.
Additionally, you can also check that the vote made it into the system and was for the correct candidate with your smartphone without compromising secrecy, so even if your computer was infected with malware, you can still make sure everything goes correctly.
See the website of the elections committee for more.