In an age of multiverses and shared universes, it was just a matter of time before the role playing game, Dungeons & Dragons, and the trading card game, Magic: The Gathering, took their immersive fantasy worlds and merged them. That's where Plane Shift: Zendikar comes into play.
MTG's Plane of Zendikar was originally conceived as an "adventure world", allowing parties of explorers to delve into ancient ruins in search of wonders and treasures, fighting the monsters they encounter on the way. Many of the Zendikar's creative roots lie in D&D, so it should be no surprise that The Art of Magic: The Gathering — Zendikar feels a lot like a D&D campaign setting book. It's got everything you need -- adventure hooks, story seeds -- and lacks only the specific rules references to adapt Zendikar's races, monsters, and adventures to a tabletop D&D campaign.
Plane Shift: Zendikar acts as a sort of supplement to The Art of Magic: The Gathering — Zendikar, designed to help you take the immersive world details and story seeds contained in that book and turn them into a D&D campaign. The easiest way to approach a D&D campaign set on Zendikar is to use the rules that D&D provides mostly as written: a druid in MTG might call on green mana and cast spells like giant growth, but she's still just a druid in the D&D rules (perhaps casting giant insect).
Plane Shift: Zendikar was made using D&D's 5th Edition rules. D&D uses a flexible system of rules designed to model any kind of fantasy world, allowing players to adapt any world if they try hard enough. The D&D magic system doesn't involve five colors of mana or a ramping-up to your most powerful spells, but the goal isn't to mirror the experience of playing MTG in your role-playing game. The point is to experience the worlds of MTG in a new way -- through the lens of the D&D rules. All you really need are races for the characters, monsters for them to face, and some ideas to build a campaign.
Finally, The Art of Magic: The Gathering — Zendikar will help you create a D&D campaign in that world, but you don't actually need the book to make use of the material in Plane Shift: Zendikar — you can also refer to the abundance of lore about this plane found on MagicTheGathering.com and the Zendikar plane profile.