You shouldn't think much of his statement about being in two places at once; it's a bunch of fluffy nonsense he's using to make his work sound sexier. A human-sized object at livable temperatures will never exhibit such behavior.
Their experiment actually used thousands of measurements to determine what was happening with the lever. I'll try to summarize, but please let me know if you need clarification.
The device they study is an electrical circuit that connects the macroscopic lever from your picture to a quantum bit (or "qubit"). At very low temperatures both the lever and the qubit are two-state systems, meaning they are individually described by two quantum states (e.g. "not vibrating" and "barely vibrating", which I'll respectively denote $\left| 0 \right\rangle$ and $\left| 1 \right\rangle$). The coupled system is found in combinations of these states, but the relevant ones in this experiment are when the qubit is barely vibrating and the lever isn't, and vice-versa. In my previous notation these states are
\left| 1 \right\rangle_q \left| 0 \right\rangle_l, \quad \left| 0 \right\rangle_q \left| 1 \right\rangle_l,
where $q$ and $l$ to refer to the qubit and the lever. After cooling down the circuit so the lever and qubit aren't vibrating ($\left| 0 \right\rangle_q \left| 0 \right\rangle_l$) they give the qubit a little energy so it is in the barely vibrating state ($\left| 1 \right\rangle_q \left| 0 \right\rangle_l$). Since the qubit and lever are connected, over time the energy moves from one to the other. Equivalently, the system evolves to a new quantum state where the lever is barely vibrating and the qubit isn't ($\left| 0 \right\rangle_q \left| 1 \right\rangle_l$). From a quantum mechanical viewpoint this evolution is attributed to the system being in a superposition of the two states
a(t) \left| 1 \right\rangle_q \left| 0 \right\rangle_l + b(t)\left| 0 \right\rangle_q \left| 1 \right\rangle_l.
The squares of the numbers $a(t)$ and $b(t)$ tell us the probability of finding the system in each of its quantum states at a certain time $t$. When $a=1$ only the qubit is vibrating, when $b=1$ only the lever is vibrating, and when $a$ and $b$ are both non-zero the lever is "simultaneously" vibrating and stationary.
O'Connell and his collaborators were able to measure the state of the qubit very accurately. They prepared the circuit in the way I described above, waited some amount of time, and looked to see if the qubit was vibrating or not. They repeated this process over and over, tallied up how often the qubit was vibrating, then calculated the probability they would find it vibrating if they performed the experiment again. For certain wait times they found that the qubit was sometimes vibrating and sometimes not vibrating. Going back to my expression for the superposition, this means that $a$ and $b$ are non-zero, or that the lever is in a superposition of stationary and vibrating states.