"The emperor sees the swineherd and the princess kiss and gets very angry."
Translation:Kejseren ser svinedrengen og prinsessen kysse, og bliver meget sur.
Without knowing the actual grammatical rule for this in Danish I can confirm that "kysse" is the correct form of the verb in this case.
I think it may have to do with what is the verb in the sentence; which is actually "ser" (sees) so the kissing part becomes something that is being seen and not the active verb? "Kysse" is the base-form of the word, i.e. "At kysse" (to kiss).