For "Land Grab", the objectives represent key locations or strategic grounds. You can imagine them as the enemy's temples or monuments. The enemy wishes to protect them, while you wish to claim them as your own. Previously, they worked as a sort of king-of-the-hill kind of objective, where each turn you held at least one of them (there were only two no matter what the army size was) the enemy would lose morale. Now that the number of objectives has changed to scale with army size and they are removed from play after being controlled for a round, you can imagine it more as your troops desecrating whatever was there. Not exactly a greedy land grab per see, but I doubt you are actually purchasing the deed to the place while in combat.
As for "Disrupt Supply Line", the objectives represent supplies for your enemy. This can be either tents, food, or even weapons. I believe the motivation represents that you are attacking the camp of the enemy. They have camp defenders, but if you are able to reach their supplies and destroy them, their army will be less effective. Nobody likes watching their tent and food burn down, leaving them without food or shelter for days or possibly weeks.
This is my understanding of the objectives at least.
