People felt that the National Bank Notes were more secure than the greenbacks that had previously been used because there were fewer banks that could issue them and because those banks were more closely regulated by the federal government.
Before the Civil War, paper money in the United States could be issued by any bank that wanted to do so. Banks issued the paper money and it was not backed by the US government. While a bill from a given bank could circulate, it only had value as long as people believed that the bank would be able to redeem it. So, if Bank X put out a $1 bill but people did not trust that bank to remain solvent, that bill might only actually be worth $.50. Some people might not even accept the bill at all. The greenbacks were like this. They were not all worth the same amount and they had not guarantee behind them other than that of the bank that issued them.
By contrast, the new National Bank Notes were backed by the federal government. The federal government chartered certain banks, allowing them to issue money. Those banks had to keep a certain amount of federal securities on hand as a guarantee. If the banks did this, they could issue money and the US government would guarantee the value of that money.
Thus, the new money was backed by the federal government while the greenbacks were backed only be individual banks. This is why people thought the new money was more secure.