I came into this thread thinking you meant requiring a percentage more than 50% to make decisions, like 70%. Then I read an answer saying that you're probably talking about an electoral system where the government needs to have more than 50% to be ratified (right?), and I have a different thing to say.
I've lived in Greece for a long time, where this is a rule. The general feeling about it is that it is common sense: a government needs to have a majority to be able to actually enact things during its mandate, otherwise the opposition would block everything and nothing would get done. Importantly, in the past 10 or so years governments have ruled with a very close margin, usually a handful of seats more than 50%, which means that if the government tries to do something particularly heinous, you do start to see dissidents who threaten to vote against (even though that means they are usually kicked off the party).
Then I moved to Sweden, where common sense is the total opposite: it is good to have minority governments, because then anything that passes through parliament needs to have broader support, and include compromises so that even some opposition parties will approve it. Otherwise the governing party can be authoritarian and pass things that quite a big chunk of the political climate in the country is against, without any compromise.
The two views above are opposite and incompatible at first glance, but I have to tell you, having lived under both, they're not fundamentally different. If you ask me the difference, I'd say that requiring a super-majority is more vulnerable to authoritarianism (which you can see currently in Greece, where the supposedly center-right government is actually doing lots of authoritarian moves (this is not a parliamentary move, but you get an idea of their style)). However, requiring a super-majority also allows for more freedom to affect things even towards the other side, ie more leftward change. With minority governments, everything that passes through parliament ends up being this wishy washy result of a tug of war, so that no matter what ideology the government has, the resulting laws always end up being super centrist, unless there's a very broad consensus in society leaning one way, allowing one side of the political spectrum to have a super-majority. One could argue that's more democratic, but I wouldn't say the difference is huge in practice.
Finally, let me write why I think that difference isn't that huge: the problem is that both systems are parliamentary and representative, so in both there's a certain class of people, career politicians, that make the decisions. In current times, those tend to be somewhat homogeneous, somewhat capitalist, and usually beholden to civility and pretexts. Recently we've seen some examples of parties or factions that are different to the classic career politicians, like Trump-era Republicans in the US, and other similar factions in other countries, and they have caused an upset in the system, but I would argue much for the worse. But regardless, old-style parliamentarians are still a huge part of the system, and I think even though it does make a difference, super-majority or not still results in more or less the same results.