Hamiltonians that are unbounded from below lead to instabilities of the physical system in question. Mathematically, it just simply means that the Hamiltonian (or crudely, energy) can take arbitrarily negative values; it has no lower bound.
There are different ways to look at this problem. Let's start classically, for a free theory. Here, it is not really a problem if you have a Hamiltonian that is unbounded from below. Things change if we consider an interacting theory. Suppose you have a system $S1$ with Hamiltonian unbounded from below and a system $S2$ with Hamiltonian bounded from below, then an interaction between these two systems will excite $S1$ to a higher-energy state, and $S2$ to a lower-energy state, without violating conservation of energy. But this can happen indefinitely, leading to an instability.
In a quantum theory, the above holds, with some modification. Even if there is no direct interaction between two different fields (one with bounded Hamiltonian, and another with unbounded), they can still be produced indefinitely from the vacuum, without violating energy or momentum conservation. Nothing prevents this process from occurring, so it must happen, leading to an instability.
Apart from appearing in the context of the spin-statistics theorem as you linked above, Hamiltonians that are unbounded from below typically appear when you have higher derivative Lagrangians/equations of motion. Read up Ostrogradksy instability. They lead to so-called ghosts, which are negative energy/negative norm states. Negative norm states, of course, do not make any sense in (probabilistic) quantum theory. Busting these ghosts from a variety of effective field theories is an active field of research.
See also my answer here on a related question and more info - https://physics.stackexchange.com/a/473778/133418