I guess this should be one of the most basic questions in Yahadut, but somehow I can’t work it out: What exactly does ’Baruch Ata Hashem’ mean? Is it wonderment at the magnificence/power of Hashem, or gratefulness for Hashem’s great deeds...?

The literal translation is "blessed are you Hashem". The commentators explain these words in different ways. Baruch – comes from the word Brecha which means – source; we proclaim that Hashem is therefore the source of all created and of all blessings. The word is also from the word Lehavrich (Gennasis 24; 11) that means – to bring down. We therefore also ask through our blessing to bring the abundance and blessing of Hashem down to us. This is by acknowledging that he is the creator and the mighty one that rules the world.