I'm the author of the slides.
It turns out the local linear trend model is closely related to a cubic spline, but the actual cubic spline result comes from a slightly different model (so there was an error in my slides, apologies).
The local linear trend model states $$ \begin{align} y[t] &= \mu[t] + \epsilon[t] \\ \mu[t+1] &= \mu[t] + \delta[t] + \eta_0[t] \\ \delta[t+1] &= \delta[t] + \eta_1[t] \\ \end{align} $$ The spline result comes from a model $$ \begin{align} y[t] &= \mu[t] + \epsilon[t] \\ \Delta^2 \mu[t] &= \eta[t] \end{align} $$ where $\Delta$ is the first difference operator. The last line is equivalent to saying that
\begin{align} \mu[t+1] = \mu[t] + \Delta \mu[t] + \eta[t] \end{align} This is awfully close but not the same as the local linear trend. The result is the discrete time equivalent of a result by Kohn and Ansley. You can find it in section 3.11 of Durbin and Koopman (first edition) or section 3.9 of their second edition. I believe that section is available in Google books previews. The reference in DK was Kohn, Ansley, Wong (1992), which you can grab if you have access to JSTOR.
For applied work, you might consider the "integrated random walk model" which is the local linear trend with the variance of eta0 set to zero. It produces smoother posterior means than the local linear trend model.
