In a Democracy, if nobody knows which candidate anyone else voted for, how do we know we can trust the results of an election?

Audit trails and exit polling.

Audit trails allow you to demonstrate that the final tally declared in the end is directly derived from the physical votes cast. This is one of the major arguments against electronic voting: electronic records can be erased and overwritten relatively easily (just look at your hard drive). You can't remove and replace paper ballots without a physical act. The ballots go somewhere, and systematically getting rid of thousands of ballots and replacing them with new ones would be much more difficult to fake than changing digital data.

Now, of course, there exist lots of countries that have exactly this problem. This is common in Africa (Zimbabwe, South Africa, Kenya) and other countries. Iraq claimed Saddam Hussein was elected with 100% of the vote in 2002. In these instances, the fraud is able to be carried out because of the oppressive and essentially autocratic regimes -- everyone knows the fraud is happening, but no one can do anything about it.

But let's assume for a second that a government does systematically change the election results. Even then, in a free country with free press, there exists a check: exit polling. News stations typically poll people as they leave a sample of polling places to see how they voted. these exit polls are how they make their predictions about who won before any votes are even counted. These exit polls are based on interactions directly between the people and the news agency -- the government is not involved. So, if a government did anything more than tilt a close election one way, it would show up as a major departure from the exit polls and prompt an investigation.

The only way for a government to systematically manipulate the results of an election would be to both manipulate the results and manipulate every single news agency reporting on the results. Given the radically different biases of the news agencies, that isn't likely to happen in most civilized countries. It's far, far cheaper and easier to manipulate elections by selectively preventing or encouraging people from voting in the first place.