I would translate it this way: Как же это? How so? Ты же знаешь, что... (But) you know that… Вы же видите, (But) you see, А как же иначе? How can it be otherwise? не разорваться же мне! I can’t do 10 different things at the same time! А как же. Yes, indeed. or Of course! I can give you some other examples, probably they could help you to understand “же” better: Он остаётся, она же уезжает — he will stay here and she will go OR He will stay here, as for her, she will go Если же вы не хотите — but if you'd rather not Почему вы ему не верите? Он же доктор — Why don't you trust him? After all, he's a doctor Когда же вы будете готовы? — whenever will you be ready? Пойдём же! — come along! Что же мне делать? — what on earth shall I do? Or….what am I to do? “же” can mean a sameness - тот же, такой же — the same так же — in the same way тогда же — at the same time там же — in the same place I think in most cases “же” is used in emphatic meaning.

no set rules you use it when you want to meow!

In Russian, particle "же" emphasizes the idea of a sentence, a speaker's intention, in fact. It is commonly used in speaking informal Russian. But I dunno if it has anything to do with Latvian. Hope that helps.