Let us construct a very simple economy with only two individuals. Let's call this world Simpletonia. These individuals, or Simpletons, will be known as Gordon and Peter.
In this world, there is only one type of resource (imaginatively named Resource) which is non-perishable and has zero storage costs. It is necessary to consume 1 unit of Resource a day in order to survive. Resource is located in one of two types of Deposits on this world - one low-yield and one high-yield - from which the Resource must be gathered before it can be consumed.
The first type of Deposit yields 2 Resources at the end of a day of continuous gathering. The second Deposit yields 1000 Resources as a lump sum at the end of a year of continuous gathering. Needless to say, each Simpleton can only gather from one deposit at a time.
It is clear that the second Deposit offers a higher return than the first (1000 Resource / 365 days = 2.7 Resource/day > 2 Resource/day). Thus, each Resource-maximizing Simpleton should gather from the high-yield Deposit as soon as possible. In a pre-debt economy, this cannot happen until each Simpleton has stockpiled enough Resource by gathering from the low-yield Deposit for a whole year. In a debt economy, Peter can immediately start gathering from the high-yield Deposit while Gordon gathers from the low-yield Deposit. This happens because instead of saving his extra unconsumed Resource in storage (where it will be a zero-yield asset), Gordon can invest in Peter by lending him 1 Resource a day.
Would you rather be indebted or debt-free? The debtor (in this example, Peter) isn't always worse off than the creditor (Gordon). Peter can probably default with little or no immediate consequence since he has no tangible assets for Gordon to seize. However, this will be sub-optimal for Peter in the long term, as Gordon will be very unlikely to lend to Peter again, and thus they will revert to the pre-debt economy in which access to the high-yield Deposit is delayed by a year.
In fact, this is very similar to the famous Iterated Prisoner's Dilemma from game theory. Just as in that game, tit-for-tat is the close-to-optimal strategy (if you cooperate -> I will cooperate in the next round, if you don't cooperate -> I will not cooperate in the next round). Thus, reputation is very important in a debt economy, which is why individuals have credit scores, corporations have credit ratings, and Argentina hasn't had access to international credit markets since they defaulted in 2002.
This is a highly simplified model, but we can see that the debt economy will do vastly better than the pre-debt economy: they have access to the high-yield Deposit, and thus more Resource, an entire year earlier. A few other things are immediately evident: debt as a whole nets to zero. How much everyone owes cannot be more than how much everyone is owed. One Simpleton's debt is another's asset. Gordon's surplus is Peter's deficit.
Even though debt always nets to zero, economic wealth isn't a zero-sum game. It is clear that Gordon and Peter will both be richer from this arrangement. Debt, in and of itself, isn't bad, because there necessarily always is someone in debt.
Finally, why is this relevant? If we rename Gordon "Government" and Peter "Private Sector" and if Gordon's surplus is Peter's deficit (as shown above), then Government surplus = Private Sector deficit. Similarly, Government deficit = Private Sector surplus. Thus, if the private sector as a whole wants to pay down its debts through generating surpluses (as it has been trying to since the 2008 financial crisis), the only way for this to happen is for the government to incur deficits. In fact, some simpletons even believe that this is justification for the government to actively pursue a policy of large deficits in order to aid in private sector deleveraging.