I read from this website that the uncertainty principle implies that seemingly empty space is filled with energy, called vacuum energy . The relevant equation I can think of is $$\Delta E\Delta t \geq \frac{\hbar}{2}.$$ Here, $\Delta E$ is standard deviation of energy for a system in quantum state $|\psi \rangle$ and $\Delta t$ is the approximate amount of time it takes for the expectation value of energy to change by a standard deviation as the system evolves.
How is this related to the statement that empty space is filled with energy?