You damage other monsters by playing attack cards, and the opponent can mitigate damage by playing defense cards as a response.
Guts aren't something you get from the cards you play, but are actually something you get by discarding cards. The core mechanic limiting the actions you can take in the game is Guts. Your goal is to defeat the opponent's team. Monsters aren't something that you summon from your deck during gameplay, but instead you use your deck to control and support your chosen monsters.
Similarly to the main video games, the Monster Rancher doesn't really the Magic the Gathering model that Pokemon uses, instead much more resembling the how the video game plays out.