Statement – I: For the planets orbiting around the sun, angular speed, linear speed and KE change with time, but angular momentum remains constant. 
Statement – II: No torque is acting on the rotating planet. So its angular momentum remains constant. 

Net torque acting on the rotating planet is zero $i.e\tau =0$

Using $\frac{dL}{dt}=\tau$

Thus $\tau =0\Rightarrow L=cons\tan t$

Hence angular momentum is constant. So reason statement is correct. 

Now $L=mvr=cons\tan t\Rightarrow vr=cons\tan t$

Thus as the distance between sun and planet changes, linear speed of the planet changes and this its kinetic energy also changes and thus its kinetic also changes. Also using $L=m{r}^{2}w=cons\tan t\Rightarrow r^{2}w=cons\tan t$

Thus as the distance between sun and planet changes, angular speed of the planet changes. Hence assertion statement is also correct and reason explains the assertion very well.