Why did money stop being backed by precious metals?
Ultimately people realized that it is waste of scarce resources to dig up gold/silver from the ground just to put it under the ground again (in a bank safe). Paper (or nowadays digital) money can fulfill all three basic functions of money (medium of exchange, unit of account, store of value) equally well and you don't waste precious metals with multiple industrial uses.
This being said even though overall the answer is that gold standard was simply too inefficient, every country will have some particular historic reason for leaving gold standard. For example, UK suspended it during the war and finally abandoned it during the Great Depression (see BOE discussion here), as gold standard can make recessions worse (I will return to this point later). In the US there were different reasons (see next section). Discussing historic reasons for every country would be beyond the scope of SE answer, but ultimate answer countries at some point decided they need some more efficient alternative.
I am trying to understand and grasp why FIAT money became so popular after Nixon 'ended' the Bretton Woods system. Why was this decision done.
The US gold standard in particular was ended due to several reasons. Federal Reserve History has good history article about it:
The U.S. share of world output decreased and so did the need for dollars, making converting those dollars to gold more desirable. The deteriorating U.S. balance of payments, combined with military spending and foreign aid, resulted in a large supply of dollars around the world. Meanwhile, the gold supply had increased only marginally. Eventually, there were more foreign-held dollars than the United States had gold. The country was vulnerable to a run on gold and there was a loss of confidence in the U.S. government’s ability to meet its obligations, thereby threatening both the dollar’s position as reserve currency and the overall Bretton Woods system.
Many efforts were made to adjust the U.S. balance of payments and to uphold the Bretton Woods system, both domestically and internationally. These were meant to be “quick fixes” until the balance of payments could readjust, but they proved to be postponing the inevitable.
In March 1961, the U.S. Treasury’s Exchange Stabilization Fund (ESF), with the Federal Reserve Bank of New York acting as its agent, began to intervene in the foreign-exchange market for the first time since World War II. The ESF buys and sells foreign exchange currency to stabilize conditions in the exchange rate market. While the interventions were successful for a time, the Treasury’s lack of resources limited its ability to mount broad dollar defense.
From 1962 until the closing of the U.S. gold window in August 1971, the Federal Reserve relied on “currency swaps” as its key mechanism for temporarily defending the U.S. gold stock. The Federal Reserve structured the reciprocal currency arrangements, or swap lines, by providing foreign central banks cover for unwanted dollar reserves, limiting the conversion of dollars to gold.
In March 1962, the Federal Reserve established its first swap line with the Bank of France and by the end of that year lines had been set up with nine central banks (Austria, Belgium, England, France, Germany, Italy, the Netherlands, Switzerland, and Canada). Altogether, the lines provided up to \$900 million equivalent in foreign exchange. What started as a small, short-term credit facility grew to be a large, intermediate-term facility until the U.S. gold window closed in August 1971. The growth and need for the swap lines signaled that they were not just a temporary fix, but a sign of a fundamental problem in the monetary system.
International efforts were also made to stem a run on gold. A run in the London gold market sent the price to \$40 an ounce on October 20, 1960, exacerbating the threat to the system. In response, the London Gold Pool was formed on November 1, 1961. The pool consisted of a group of eight central banks (Great Britain, West Germany, Switzerland, the Netherlands, Belgium, Italy, France, and the United States). In order to keep the price of gold at $35 an ounce, the group agreed to pool gold reserves to intervene in the London gold market in order to maintain the Bretton Woods system. The pool was successful for six years until another gold crisis ensued. The British pound sterling devalued and another run on gold occurred, and France withdrew from the pool. The pool collapsed in March 1968.
At that time the seven remaining members of the London Gold Pool (Great Britain, West Germany, Switzerland, the Netherlands, Belgium, Italy, and the United States) agreed to formulate a two-tiered system. The central banks agreed to use their gold only in settling international debts and to not sell monetary gold on the private market. The two-tier system was in place until the U.S. gold window closed in 1971.
These efforts of the global financial community proved to be temporary fixes to a broader structural problem with the Bretton Woods system. The structural problem, which has been called the “Triffin dilemma,” occurs when a country issues a global reserve currency (in this case, the United States) because of its global importance as a medium of exchange. The stability of that currency, however, comes into question when the country is persistently running current account deficits to fulfill that supply. As the current account deficits accumulate, the reserve currency becomes less desirable and its position as a reserve currency is threatened.
Which cons and pros did it had?
When multiple countries are on gold standard or if the exchange rates are fixed with respect to currency that is on gold standard (e.g. Bretton Woods system with other currencies having more or less fixed exchange rate vis-à-vis dollar) they have in essence fixed exchange rate vis-à-vis each other. That eliminates exchange rate risk between trading partners.
Gold standard makes it harder (although not impossible) for government to expand money supply (see BOE). Historically, governments were often engaging in excessive monetary expansions ('money printing') to pay for large debts which often led to high levels of inflation and financial instability (see Reinhart and Rogoff 2009). However, in present day most fiat systems are administered by technocratic central banks like ECB and Fed often by Ivy league PhD economists where political influence is minimal. So this is less of an issue in modern era (even though Ivy league economists still can and sometimes do make mistakes like Fed and ECB probably being too aggressive during Covid19 crisis).
Gold standard makes crises worse. The de facto fixed exchange rate when countries are at gold is both advantage and disadvantage. If there are asymmetric shocks (one country experiences recession and other doesn't) fixed exchange rate makes much more difficult to accommodate these shocks (between fiat currencies exchange rate just moves to accommodate asymmetric shocks). Even domestically it makes crises more difficult to deal with because one potential way how to mitigate crisis is precisely through monetary expansions and gold standard makes that harder.
Gold standard makes sustaining large government spending and debt difficult. When gold standard was first adopted most governments would spend about 3-5-10% of GDP. Modern day median for OECD country is about 48% (see OECD data). This massive spending can be handled more easily when government can monetize some of this spending through monetary expansion.
As mentioned at the beginning, gold standard is quite inefficient. You dig up gold from ground and instead of using it for industrial purposes you just put it in an underground vault to sit there unproductively. It makes sense to use some material that is not as scarce. Paper (or nowadays digital bits) can be used as a medium of exchange, unit of account and also store of value. Although, historically gold was better store of value than fiat money, this was due to policy choices of government. In principle, even with paper money government can keep inflation arbitrarily low (but that is not always optimal for economy).