What is implied by the saying "Eternal vigilance is the price of liberty?"
The main implication of this saying is that there will always be some person or some group that is trying to take our liberty away. In the United States today, most people would tend to say that it is the government that is doing this.
The quote is saying that we must always be vigilant, we must always be on the alert, if we want to keep our liberty. If that is the case, then there must be someone out there who is always trying to take that liberty away. The most plausible candidate is the government, since it has the power to do so. Therefore, the saying implies that someone (most likely the government) is always trying to take our liberty and we must therefore always be on our guard.