At its most basic, "inflate or die" means that given the massive amounts of debt that the United States has created, if they stop printing money (the main way they can cause inflation) the system will collapse. I found a very clearly written article with interesting graphs on how this came to be at this site. A key paragraph:
It is simply not possible to grow your debts faster than your income forever. However, that’s been the practice since 1980, and current politicians and Federal Reserve officials developed their opinions about “how the world works” during the 33-year period between 1980 and 2013.
Put bluntly, they want to get us back on that same track, and as soon as possible. The reason? Because every major power center, be that in D.C. or on Wall Street, tuned their thinking, systems, and sense of entitlement, during that period. And, frankly, a huge number of financial firms and political careers will melt away if and when that credit expansion finally stops. And stop it will; that’s just a mathematical certainty.
It's important to remember the human factor when discussing economics, as personal motivations can be as telling as mathematical models. One key to understanding "inflate or ide" is this: a huge number of financial firms and political careers will melt away if and when that credit expansion finally stops. The average person has personal debts and no real motivation to keep the debt cycle moving forward, but those at the top of the pyramid, who have created a system where every household holds large debts, have the most to lose if the system collapses. Thus, they have every incentive to pursue absurd monetary policies where inflation and money printing are sold to the public as a good thing, and not what it really is-- kicking the can down the road for future generations to deal with when the whole house of cards collapses.
Perversely, as market debt grows, so do the incentives for those at the top to keep inflating and keep people borrowing. A recent example, shown in the graph I've attached to this answer, shows the insanity of how a small market contraction almost caused the system to collapse in 2008-2009, which was rescued by increased money printing, sending debt back up to "normal" levels. If this sounds a bit like a Ponzi scheme, that's because it is.
Russell's point in the "inflate or die" setting is to highlight what he saw as a primary focus of Fed policy. Russell argued that the Fed wanted to avoid a deflationary economic setting. Deflation is something he sees as a decrease in the prices of goods within the marketplace setting. Russell suggests that to avert the idea that the United States "might be following the deflationary path of Japan," the policy of the Fed is to "inflate or die." Russell sees this policy in a couple of distinct areas. One such area is ensuring that commodity indices remain high, on the inflationary side. Russell sees this in how home prices are holding their prices at a high level and how Fed intervention prevents prices from dropping. Intervention in the gold prices to ensure that it remains at a particularly high point is another example of how the Fed believes that maintaining prices on the high side is essential to Fed policy development. This trend of institutional buying to ensure high values on commodities is a reflection of the Fed's tendency to embrace the stance of "inflate or die." To ensure a high standard of prices and services of commodities is the basis of what Russell sees as "inflate or die" in Federal Reserve policy.