Stated reasons were (1) reducing load on health services from road accidents (which has worked - my friend in A&E says he's "never sat around on his arse so much at work before"); (2) reduce transmission of virus at fuel stations, risk to breakdown services, etc
If I remember rightly, there was actually a bit of a controversy over this because the police in some areas got it into their heads that no-one should be driving to places to walk full stop and started pushing that message and complaining about people not following it, even though that wasn't actually a rule. Though it's entirely possible a lot of the media coverage of that has also been memory-holed and only exists in the Wayback machine; the BBC in particular have been completely rewriting articles a lot lately.
(The restriction to one session of exercise per day was also only legally enforceable in Wales and not England, for some reason, even though it was part of the advice in both.)
There was a law mismatch here. The government were saying you can't do that, however the legal powers they had (and the powers they delegated to the police) were not sufficient to actually stop people for some of the things they were stopping them for.
I think Derbyshire police were the worst for that.
So it's possible here that Dominic Cummings didn't break the lockdown laws, but he certainly broke the lockdown rules his own government were telling the public.
I feel for people stuck in apartments!