The trust that money has value gives it its value. If money can be exchanged for gold, then this makes people trust that it will have value. If this is not the case (and it is not, although it used to be the case almost everywhere) then the only thing that can make people trust that money will not lose its value through extreme printing of money is having trustworthy people endowed with the power to print money. These people are called central bankers, hence their credibility matters here. The consequences of losing faith would be (hyper-)inflation.
First a few definitions. The purchasing power of money is basically its value. That is how many goods/services (e.g. bread) you can get for a dollar. Inflation is a price increase, which means that a dollar can now buy less things (as they cost more). Hence money loses its purchasing power when inflation occurs.
Further, the value of money is basically determined just like the value of anything else - supply and demand. So if you print a lot of money (supply increases) its value will go down. That's the reason the government doesn't just print a bunch of dollars and give them to everyone, because that money would lose value, i.e. it would lead to inflation (rising prices).
The reason that money has any value at all in the first place (today) is because people believe it has value. Also, people believe it has value, because it does. They observe that they can use it to obtain goods and services. That's pretty much the only thing distinguishing the dollar from monopoly money today. As is apparent from the reason money has value it is easy to get into vicious cycles. If people stop believing it has value then it won't, which will lead to more people believing it doesn't and so on.
This is why hyperinflation is so hard to stop. Once money loses a lot of its value the trust people have in this money is undermined. They panic and think it won't be worth anything soon. So they try to get rid of it by exchanging it for other things. As everyone tries to exchange money for other things, effectively the supply of money (really the velocity of money) increases, which undermines its value even more. Since everyone wants to get rid of money in this scenario, there is no one to give it to in order to trade with so it becomes worthless (as no one wants it). A german central banker once said that inflation is like toothpaste: once you get it out of the tube it's very hard to get back in again. The problem is, that governments have an incentive to print more money, to pay off debts and have more spending power, etc., which creates inflation.
So there are multiple equilibria possible, where all people (or a critical mass) believe in money and since all do there is no reason not to believe in it. Or nobody does and in this case there is no reason to in fact believe in money. It's also possible for many to believe in it and many not to (e.g. 50-50 or 60-40), but this is not a very stable equilibrium. For more on how money comes into existence at all, see the paper by Kiyotaki and Wright ("Money as a medium of exchange" and "A Search-Theoretic Approach to Monetary Economics" american economic review 1993) , who discuss these equilibria possibilites. There, basically money emerges, because barter (trading goods for other goods) is so inefficient as it requires a rarely occuring double-coincidence of wants. So if money is undermined, probably something else will emerge. E.g. in Germany after a crash people used cigarettes as currency.
Now it wasn't always the case that money only had value, because others believed it. On top of that, governments had what was called the "gold-standard". This meant that at any time, money could be exchanged for a certain amount of gold. Because of this, governments were restricted in how much money they could print, by how much gold they had in reserve. So this was meant to stopp excessive printing of money and to stop inflation panicks. This however came at a cost, because monetary policy could barely be used and it required governments around the world to frantically search for gold. Nowadays, we don't have the gold standard, so governments are free to do what they want. However, they don't abuse their power, which is evidenced by low inflation rates in the last 30 years and without the drawbacks of the gold standard.
If the central bank is not credible and the government does not guarantee to exchange money for gold, basically there is nothing to "anchor" the value of money. So if people expect the central bank to print too much money, then money will lose its value.
One last possibility through which the mistrust of the central bank is important here is because of wage-setting. When negotiating wages, unions take into account rising prices (inflation) so that they don't become poorer through price increases, which is why they demand higher wages. If they expect a lot of inflation to occur in the course of the year, they will demand high wages right now (as these contracts cannot be renegotiated every month). However the rising wages will lead to higher prices, which in fact is inflation and this spiral could get out of control. With a credible central bank, this won't happen of course, as the expectations aren't so wild.