Why does the IMF push for austerity? Are there any countries that applied austerity measures and had a positive outcome that was attributed to it? Why do institutions like the IMF often encourage or require it?
Why do so many people die in hospitals? Aren't doctors there supoosed to save lives?

The IMF is the trauma surgeon of government finances. When a country asks the IMF for help, it is because its finances are in very bad shape. They no longer have enough money to pay for public services, infrastructure, pensions, salaries. Without some kind of external help, the government has to fix the problem by some combination of reduced spending, increased taxes, or by printing money (which will increase inflation).

What this shows is that austerity is not imposed by the IMF. Austerity is "imposed" by the inability of the government to finance its own spending.

The IMF, by providing loans when no one else will, actually helps to soften the adjustment. With the extra funding, the government is able to cut spending and raise taxes less than they otherwise would.

Of course, one thing to keep in mind is that the IMF is not a charity. It expects its money back. If the IMF provided grants instead of loans, there would be no need for austerity. But then the IMF wouldn't exist, because it would have run out of money long ago.