The original question here asked the meaning of “hold the fine.” I have edited it to read “hold the line” because I am certain that that is the correct phrase. This phrase means that taxpayers expect elected officials to prevent the agencies’ budgets from getting bigger.
In America today, many people think that government has gotten to be too big. They think that there are too many government agencies and that those agencies spend too much money. Therefore, they do not want those agencies’ budgets to increase. The people who can prevent budget increases are elected officials who have control over the budgets. Therefore, the taxpayers expect them to prevent increases in the size of agencies’ budgets.
The idiom “hold the line” means to stand on one’s principles and not allow something to change for the worse. The link below argues that the idiom comes to use from American football, where it refers to a defensive team trying to prevent the offense from pushing down the field. Whether the idiom comes from sports or from the military (referring to soldiers needing to hold their positions), the meaning is still the same. This idiom means to prevent something from changing for the worse. In this context, it means that taxpayers want elected officials to prevent agency budgets from increasing because they feel that budget increases are a bad thing.