What's the idea behind MTG?

I'm still new to MTG and learning more every day. I love the lore and art of the setting most of all so far but I still find myself a bit confused about the main idea behind what MTG is supposed to be so I'm hoping one of you can help me understand.

What do I mean?

Well for example in Pokémon I know the overall idea. You're a Pokémon trainer. You catch, train and battle your Pokémon against other trainers and hope to win gym badges and complete the Pokédex. It's a world where Pokémon are strange animals with elemental powers and near-human intelligence. We bond with them etc.

But what's the equivalent idea for MTG? It's a multiverse of fantasy worlds ok. But are we, the player, supposed to be a planeswalker summoning various characters from these worlds to do battle with one another? Are these various worlds supposed to have some kind of connection and/or contact with one another outside of the card game or are they strictly separate worlds with separate lore? Anything more to add? I'd love to hear! Thanks a lot to anyone who takes the time to read/reply!