How did redenomination prevent the Weimar Republic from further decline?

The previous answer is not quite correct.

The hyperinflation was stopped by the introduction of the Rentenmark and not by the Reichsmark which was introduced about 11 months later.

The problem was that the paper money was not backed by anything that peaple would trust.

Around August 1923 the plan was developed to mortgage state owned property based on its prewar gold value.

Banknotes would then be issued that would not exceed 3.2 billion gold marks at a ratio of 1:4.20 US Dollar.

Since property had a constance value and the amount issued was stricly restricted, the hope was that the population would trust the issued paper money enough to stop speculation and bring the inflation to a halt. Some literature of the time expressed surprise that this actually worked.

By July 1924 most of the official old paper money has been converted to Rentenmark. Emergency notes, issued by private organizations, had to redeem the notes themselfs.

Starting September 1924, Reichsmarks were slowly introduced that started to replace the Rentenmark, but always taking into consideration that resposible budget was to be maintained so that the confidence in the currency would not be lost.

The Reichsmark was considered an internal currency which was not freely convertible and the export of which was strictly restricted. In many ways it was similar in nature to sterling in the 1950's and 1960s.

For those interested in the details Die Reichsbank 1901 1925 is a good source published by the Reichbank in 1925 (in German) . A pdf can be downloaded from the main page of the given link.

You might want to take a look at https://en.wikipedia.org/wiki/Hyperinflation_in_the_Weimar_Republic#Stabilization and the references there (one of them, "The Rentenmark Miracle," has an outdated URL - the correct one is http://www.economia.puc-rio.br/gfranco/Ch10.PDF). In short, it wasn't just redenomination, it was (among other things) that a new currency was issued that was indexed to something of real value.

It did not save Weimar Republic, it just erased people's savings

Before the war, German mark was on gold standard. This means that you could exchange your paper banknotes for gold at fixed price (fixed exchange ratio), therefore banknotes were "as good as gold" and were used for savings as more practical and only legal tender.

Now, war cost money, and wars also ruin economy. Especially large wars like WW1. No country participating in WW1 could sustain costs of the war, so they all started borrowing hoping to repay debt by spoils of war. Borrowing happened in two ways. First, large banks and other wealthy creditors would voluntarily buy war bonds. Banks did knew there was slim chance they would get paid in full in gold (even if the war is won) , but they hoped they would hedge their loses and control large part of the economy when dust settles, as repaying of government bonds is usually given highest priority. Second, involuntary borrowing from general public, happened simply by removing gold standard and introduction of so called paper marks, which by the nature of it all immediately started loosing they worth, forcing the government to issue more of them and effectively eating away savings by inflation.

As we all know, Germany lost WW1, therefore not only there were no gains from which to repay debts, they also had to pay harsh war reparations. As mark started losing value, German government started printing more of them (to buy foreign currency and gold, to pay workers etc ...) . On the street mark became worthless, but nominally there was still promise that one day paper marks would again be valued same as pre-war gold mark. To stop all of this madness, Reichsmark was introduced, valued as one trillion of paper marks. Effect of this was that all old savings were practically wiped out . Debt holders, except those which were holding bonds explicitly valued in gold marks, were also wiped out.

Effects of this decision were some form of price stability, but also enormous impoverishment of German citizens, especially working class. They were now working for much lesser wages then before the war, and as said before all of their savings were wiped. German industrialists did benefit somewhat, as their products became more competitive on foreign markets do to lower labor costs. Large creditors and countries demanding reparations also benefited somewhat as their demands were made in gold marks i.e. certain amount of gold, so they did not lose value, and German economy was able to repay some of these.

But overall effects on Weimar Republic were catastrophic. Impoverishment and rising wealth inequality spurred rise of Communists and National Socialists, as more and more Germans became disillusioned with so called liberal democracy. There was a rise of prostitution and others forms of decadency, especially in large cities like Berlin. Finally, with Depression and German banking crisis in 1931, caused by foreign money withdrawing from German banks, Germany as a nation was ready for "firm hand" and militarization.