The levels indicate how much provocation is required before the sentinels will attack. Sentinels are provoked by killing animals, mining resources, or destroying reinforced objects or resource caches.
For example, on more permissive planets, you can kill an animal and get away with it, even if a sentinel sees. On stricter planets, they can get provoked by simply mining a large number of resources in front of them.
I believe it is also linked to how strong of a response you get once they begin attacking you. A stronger response can mean both a greater number of attacking sentinels and deadlier sentinels. There are powerful, ground-based sentinels that will show up when they get really mad.
On planets with the highest levels of activity (like Aggressive or Frenzied), your presence is provocation enough. Sentinels will attack on sight and immediately call in support.
It's worth noting that there is a correlation between sentinel activity and valuable resources. The sentinels' role is to protect nature, and what the player considers "rare", they consider "endangered".