Euro perspective - When I hear fiscally conservative, that means supporting a governmental policy that is frugal with spending and responsible with public assets and finances.
This has several parts, here are some of the most important:
a) Keeping a balanced budget - the government should not be spending more than it is collecting from taxes and income. (A little debt in dire times is fine, but that should be payed off when possible)
b) Responsible management and long term planning - the planning horizon should be counted in decades
c) Focusing on core tasks: national security, infrastructure, healthcare, education etc.
d) Not raising taxes unless strictly necessary, lowering them if it is permissible according to the above.
Socially liberal => supports personal liberties
Now why does government debt even matter? Well, because debt is owed somewhere, and if it becomes large may mean that the government is beholden to other parties for the debt.
Usually fiscal irresponsibility comes in the form of lavish promises (subsidies, tax cuts, projects etc.) with a jarring absence of an answer to the question "How are we going to finance this?"
Right, the entire phrase, at least in the US, is intended to short circuit the broader conversion about roles and mechanisms of government. It's begging the question. What US conservatives call "fiscal responsibility" typically just means "I don't think the government should spend money on things which don't personally benefit me." Or even "I want other people's children to pay for my entitlements."