It's actually a genitive case here (which basically has the same form as the dative case though when used with a noun). You can see this is actually a genitive case, rather than a dative one, when using the pronouns - (Genitive) Câinele sare asupra mea/ta/lui/ei/noastră/voastră/lor VS (Dative) Grație/Datorită mie/ție/lui/ei/nouă/vouă/lor.

When you translate this to English, you have to use 'on' to make it more natural. But if you want to make a more literal translation it would be 'on top of', I guess. On top of me/you/him, being the 'of' part the closest you can get in this case to a genitive case in the English phrase.

Perhaps this example is clearer: ”Câinele sare asupra masei” - The dog jumps on the table. ”On top of the table” - the romanian sentence is actually that the dog jumps to the upper part/ the top part of something, whose upper part / the table's.

I hope my explanation made it clearer to you, rather than more confusing...

Like all prepositions in this lesson, they require the Dative case. Typically in English you would use "on".

Or "of"