Here is a full explanation. I am changing the notation $\hat{e_r} \to \hat{r}$, $\hat{e_\theta} \to \hat{\theta}$.
We know that $\textbf{R} = r \hat{r}$. Hence, $\textbf{V} = \dot{r} \hat{r} + r \dot{\theta} \hat{\theta}$. Let us also denote by $\textbf{V}r = \dot{r} \hat{r} (= \textbf{V} \cdot \hat{r} \hat{r})$, and $\textbf{V}\theta = r\dot{\theta} \hat{\theta} (= \textbf{V} \cdot \hat{\theta} \hat{\theta})$. 804
It is clear (I hope so) why is that- the change of $\textbf{R}$ is attributed to the change along $\hat{r}$ (which is \dot{r}), and by the change of the direction of $\hat{r}$, which is $r\dot{\theta}$.
Deriving this, we see that $\textbf{a} = (\ddot{r} - r \dot{\theta}^2) \hat{r} + (r\ddot{\theta} + 2 \dot{r} \dot{\theta}^2) \hat{\theta}$.
Now for the acceleration, Let us split it into the change of $\textbf{V}r$ and of $\textbf{V}\theta$ (and then the change of $\textbf{V}$ will be their sum, and it is the acceleration).
Let us look first at $\textbf{V}_r$:
We can see that $\Delta \textbf{V}_r$ is the vector sum of the change along $\hat{r}$ direction which is $\Delta \dot{r}$, and the change caused by the change of direction of $\hat{r}$, which is in the direction of $\hat{\theta}$, with a length which is approximately the length $|\textbf{V}_r| \cdot \Delta \theta = \dot{r} \Delta \theta$ of the arc in $\hat{\theta}$'s direction. Overall we get $\Delta \textbf{V}_r = \Delta \dot{r} \hat{r} + \dot{r} \Delta \theta \hat{\theta}$ with $\Delta t$, $\Delta \theta$ infinitely small, therefore with dividing by $\Delta t$, in the limit we get $\dot{\textbf{V}_r} = \ddot{r} \hat{r} + \dot{r} \dot{\theta} \hat{\theta}$.
Now for the more interesting $\Delta \textbf{V}_\theta$:
We can see that $\Delta \textbf{V}\theta$ is the vector sum of the change caused by the change of direction of $\hat{\theta}$, and the change along $\hat{\theta}$. The change caused by the change of direction of $\hat{\theta}$ is the approximately an arc in the direction of $-\hat{r}$ in with length $|\textbf{V}\theta| \cdot \Delta \theta = r \cdot{\theta} \Delta \theta \cdot (- \hat{r})$.
The change along $\hat{\theta}$ is of length $\Delta (r \dot{\theta}) = \Delta r \dot{\theta} + r \Delta \dot{\theta}$, which can be explained as the cange caused by the increase/decrease in the angular velocity, summed with the change caused when we increase/decrease the length $r$, which causes the velocity along the arc (which is of length $r \dot{\theta}$ to increase/decrease to be $(r + \Delta r) \dot{\theta}$. With deviding by $\Delta t$ and looking at the limit, we get $\dot{\textbf{V}_\theta} = - r \dot{\theta}^2 \hat{r} + \dot{r} \dot{\theta} \hat{\theta} + r \ddot{\theta} \hat{\theta}$.
Overall we get the equation we needed. 804