$p(0)\geq c'(0)$ is assumed so that the inverse demand curve originates above the marginal cost curve.
But what is the intuition behind this assumption? The highest willing-to-pay consumer is more than the marginal cost of producing the first unit? Is this going to be the variable cost of producing the first unit?
You intuition about the assumption is correct.
Given the usual assumptions of downward sloping demand curve ($p'(q)<0$) and non-decreasing supply curve ($c''(q)\ge0$), if $p(0)<c'(0)$, then demand would lie entirely below supply. This means the equilibrium quantity must be zero---as if the market did not exist. Therefore, for the analysis to be non-trivial, the assumption of $p(0)\ge c'(0)$ is maintained.
With the use of calculus, we are assuming implicitly that quantities are infinitely divisible. Thus $c'(0)$ can be roughly interpreted as the cost of producing the "first infinitesimal unit" of the good. Mathematically, \begin{equation} c'(0)=\frac{\mathrm d\, TC(q)}{\mathrm d\,q}\Bigg\vert_{q=0}=\frac{\mathrm d\,(FC+VC(q))}{\mathrm d\,q}\Bigg\vert_{q=0}=\frac{\mathrm d\,VC(q)}{\mathrm d\,q}\Bigg\vert_{q=0}. \end{equation} Thus $c'(0)$ is the derivative of the variable cost function, evaluated at $q=0$.
