Is there an intuitive explanation for what complex basebanding a signal is?
What exactly is the complex baseband of a signal? I sort of understand what &quot;baseband&quot; means, but where does the complex phase come from?

A baseband signal is one where the center frequency is $\approx 0$ Hertz. There may be some small offset, but the idea is that it is no where near a passband signal whose center frequency is in the millions or billions of Hertz.
For a real signal, the frequency spectrum is symmetric. Due to the up-conversion process (converting baseband to passband so that the center frequency is no longer $0$ Hertz), we can put data on two orthogonal carriers, $\text{cos}$ and $\text{sin}$, in the passband signal:
\begin{align}
x(t)&amp;=\text{Re}\bigg(s(t)e^{j2\pi f_ct} \bigg) \\
&amp;= x_I(t)\text{cos}\big(2\pi f_ct\big) - x_Q(t)\text{sin}\big(2\pi f_ct \big)
\end{align}
where $s(t)=x_I(t) + jx_Q(t)$, is the complex baseband signal. For a nice explanation of what is probably happening in your MATLAB function on how to convert passband to baseband, check out this previous answer: https://dsp.stackexchange.com/a/43281/31316.

Related, does the phase have any relation to the signal? If I input a sinusoid, what should I expect the phase output to be?

There is a relation...it is the phase, that is all. The phase is helpful though and can help to pack more bits of information into a single symbol.