To answer your last question first (see detail later): the proportional hazards assumption (PH assumption) is about whether the impact of a particular risk factor might be constant over time, irrespective of the functional form of that risk factor (categorical, linear, splined function)
Natural cubic splines (more commonly called restricted cubic splines in my experience in biostatistics - this is also the terminology used in the
rms package and might help you find other relevant material) are instead about how the hazard function for the outcome (incident case of Type 2 diabetes) differs across another continuous dimension, in your case the range of RBP4 values (ranging from around 20 to 90-ish).
The spline function effectively allows for a non-linear relationship between the risk factor and the outcome (the u-shaped curve). You can see from the figure that those with levels of RBP4 either below or above the reference level of 38 units have an elevated hazard ratio relative to the reference.
But this is about the
risk factor -> outcome relationship, rather than the PH assumption. The splined function is still assumed to meet the proportional hazards assumption. It is modelled as a constant effect over time -- that is, low and high RBP4 values are associated with a higher instantaneous risk of incident diabetes, but this relative difference in hazards has been modelled as constant across the follow-up period (which is the proportional hazards assumption).
As a final note: Assessing the proportional hazards assumption is relatively simple for a categorical predictor variable (e.g. using log-log Survival curves, usually presented as stratified curves for a categorical factor). This process is considerably more complex for a continuous covariate (whether modelled with splines or with a linear function), and would really warrant its own question (if you're interested, and which I'd leave for someone else to answer!)