$\newcommand{\pr}{\mathbf{Pr}}$
Another possible intuitive interpretation would be: this means that the behavior of $D$ does not change perceptively. Suppose $D$ outputs 0 or 1. Then, the output behavior of $D$ can be summarized by the probability distribution of $D$'s output, and this can be written as a vector $(\pr[D=0], \pr[D=1])$.
Consider the L1 distance between the distribution vectors with respect to the oracles $O_0, O_1$:
$$|\pr[D^{O_1}=0]-\pr[D^{O_0}=0]|+|\pr[D^{O_1}=1]-\pr[D^{O_0}=1]|.$$
Since $\pr[D^{O_1}=0]=1-\pr[D^{O_1}=1]$ and $\pr[D^{O_0}=0]=1-\pr[D^{O_0}=1]$, plugging these into the L1 distance, we get
$$2|\pr[D^{O_1}=1]-\pr[D^{O_0}=1]|.$$
So, the given condition is that the output probability distribution doesn't change much, when one swaps $O_0$ and $O_1$.
This makes sense: to be able to distinguish two situations means to be able to act differently according to the given situation. If someone's behavior never changes (and cannot change) even if you switch one from the other, it means he/she doesn't recognize the switch.
What if $D$ outputs not only a bit but something more? Say, $D$ could output a number. Even in that case, if $D$ behaves differently when the oracle is switched, some aspect of the output must change. For example, say, the MSB of the output of $D$ could change. In that case, we may define another distinguisher $D'$ which runs $D$, and then only outputs the MSB of $D$'s output. So, if there's no such $D'$, then there's no such $D$. So more or less without loss of generality, we may consider only distinguishers with binary output when defining indistinguishability.
