Questions about the gameplay of Heroes 6.

And as soon as I opened the bridge where the enemy hero is located, he started moving, purposefully collecting resources and approaching my cities.

Most likely, this is just a coincidence. The only thing you can provoke the enemy hero to do is to attack. He may patrol a certain territory and attack everyone he encounters.
It is also possible that the enemy simply hired a new hero, sensing that you are approaching his city. And since he hired one, why not collect resources nearby. Especially if your hero does not pose a significant threat to him.
Also, some event that you committed may cause the AI to act in a certain way. For example, completing some mission objectives, etc.

Guys, can you tell me what the Staff of Oblivion is (it can be bought with "premium" currency)?
They also offer to buy a hero, it seems like Wiera, is it worth buying these things or is it better to find everything in the game?

In principle, it's a good staff if you fully upgrade it. It enhances some of the hero's characteristics (as do all weapons), and it takes away the selected monster's special abilities for several turns. Moreover, as far as I know, any abilities. Both the spells he casts and various cavalry bonuses, jumping across the entire field, setting traps, etc. And it seems that at level 5, it also adds a stunning effect (-10 initiative) to the hero's attack spells. You won't find this staff anywhere else in the game. I bought it without hesitation. As for the hero, I don't know. I'm not in a hurry to spend points on him yet.