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.
That any entity that becomes overburdened with debt, should cut back on its expenses, is so self-evident it does not need explaining. So "austerity" (if that is what the OP means by the word) need not be required by anybody in particular, it is self understood the moment one starts to talk with "lenders of last resort" like the IMF is.
The fact that in many cases, these "austerity measures" appear to not lead to a stroner economy (which is something bigger than just the public sector), has to do not with the imposition of austerity itself, but of the specific mix of measures taken in order to reduce public spending.
I used to think that IMF was so "enslaved" to "Supply side economics", that it was indeed the one responsible for the wrong mix of measures: demand doesn't matter, what drives the economy is production. So cut wages and raise indirect taxes - but decrease income tax to provide incentives for investments... and the like (a recent failure on the matter was Japan, see this post).
But after experiencing the crisis in my country (Greece), I came to appreciate that IMF does not go as far as substituting for the Government. What this means? It means that to a respectable degree, the measures decided and taken are determined by what the political system (and the people) of a country are willing to accept.
And the thing is, a public sector overburdened with debt is without exception a public sector larger than what the domestic economy can sustain. But this also implies that the economy, and the society, have come to depend on the public sector more than they should (given the wealth and the income generating potential of the country). So when it comes to cut public sending, they (the locals) do not want to see this happening by reducing the size of the public sector. So "reducing the size" becomes the secondary strategy and "reducing expenses" becomes the primary strategy -the domestic resistance is too great even for a powerful lender to not take it into account, when setting its conditions for the loan.
But this leaves no option but to propose and arrive at measures like wages cuts etc, hurting demand, and worsening the recession that has set in as the public sector (on which the economy depends to an unhealthy degree) went into debt crisis.
It should be the other way around: shrink decisively the size of the public sector by transferring activities (and employees) to the private sector, and do it fast. In this way disposable income will be reduced less (or not at all), and tax increases may not even be needed... but I doubt there will ever be any domestic consensus on such a policy, in a society that has come to consider a large public sector as a self-understood "constant" in its life.
I do not endorse the stance that there is an unconditional "small is better" dictum for the public sector of an economy. What I do endorse, is that "the size of the public sector should be no bigger than what the economy can sustain". It is in this relative sense that a public sector can become "overblown". And then it is obvious that the problem (in quantitative terms, which is what matters) is not "wasteful expense" but the size of public sector itself.
You can categorize their reasons for doing so in many ways, depending on each scenario. Here are a few examples:
Note: 4) is hotly debated.
Attempt to control bond yields was an important issue for Greece, and one that the average voter in a referendum could not be expected to understand. The other points are self explanatory, although the evidence for them are surely not scientific by any standard.