How to intuit a necessary optimality conidtion involving growth rate of Lagrange multiplier in Ramsey growth model?
The multiplier is equal to marginal utility of consumption. In case of CRRA utility (i.e. $\lambda_t = c_t^{-\gamma}$), its rate of growth is then related to consumption growth:

$$
\frac{\dot \lambda}{\lambda} = \frac{\mathrm{d} \log \lambda}{\mathrm{d} t} = \frac{\mathrm{d} (-\gamma \log c)}{\mathrm{d} t} =  -\gamma \frac{\dot c}{c}
$$

The condition can be then rewritten as

$$
\frac{\dot c}{c} = \frac{1}{\gamma} \left[ (f'(k) - \delta - \xi) - \nu\right],
$$
so that the consumption growth rate is proportional to return on capital (less rate of time preference), which is pretty much the same as Euler equation in discrete time.