Why are different cases of the personal pronoun use in the two parts of the sentence "si enim amici me in caelo videbunt, omnes tandem mihi credent" and what do they mean?

There are two pronouns in this sentence, 'me' and 'mihi', which both translate to 'me' in English. me is accusative, while mihi is dative. This is because of the roles they play the different parts of the sentence. me is the object of the verb videbunt, and this verb takes its object in the accusative case. mihi is the object of credent, one of the few verbs in the GCSE lists which takes a dative as its object. Both are 1st person (I/me) pronouns, meaning that they refer to the speaker.
This is a conditional sentence, where the protasis (the 'if-clause') comes first (si enim amici me in caelo videbunt), followed by the resulting apodasis (the 'then-clause') afterwards (omnes tandem mihi credent). You could translate it as 'if, then, my friends saw me in the sky, they would all at last believe me'. Here you can see how the two different pronoun forms fit into the English translation. I hope that helps! :)