In short Asset Backed Currency is currency (paper) whose value is based upon the assigned value by the country issuing the paper based upon the sovereign assets which the country owns.
The assets that could be used could be anything of market value. History had the United States currency before Roosevelt and then Nixon backed by gold. In fact, up and until Roosevelt ceased the process of owning gold and confiscated all the gold from US citizens you could take your paper money and go to any bank and exchange it for gold.
The current words of asset backed currency could be interchanged with words like sovereign currency, sound money, or anything else that defines the money to be issued backed by asset value.
As we move through the world restructure each country will issue their own currency. You may say well they do that now and your would be correct. However, the difference between today and what will be is that this currency issued by each country will not only be used in-country, as is the case today, but also as international currency. This means that every country can use their own currency in any other country to buy and sell products goods and services.
Today the world uses two currencies. One currency, except Europe and they use the EURO, for in country usage of its citizens and a separate currency used for the settling between countries on international trade. Currently the currency used to settle all international trade is the US Petro Dollar.
This means that in the US we have two currencies. One that the US citizens use " The Federal Reserve Note" and one that is used to settle all trade transactions between the US and any other country "The Petro Dollar".
So in order to move into the ability for the world to issue their own currency and use it for domestic (in-country) and international use all countries must come off the usage of the Petro Dollar first.