Dungeons & Dragons and Magic: The Gathering are two different games, but that doesn't mean their Multiverses can't meet.
From the beginning, Magic's plane of Zendikar was conceived as an "adventure world" where parties of explorers delve into ancient ruins in search of wonders and treasures, fighting the monsters they encounter on the way. Many of the plane'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 littered with adventure hooks and story seeds, and lacks only the specific rules references you'd need to adapt Zendikar's races, monsters, and adventures to a tabletop D&D campaign. And it's all surrounded by amazing fantasy art that holds boundless inspiration in itself.
You can think of Plane Shift: Zendikar as a sort of supplement to The Art of Magic: The Gathering—Zendikar, designed to help you take the world details and story seeds contained in that book and turn them into an exciting 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 on Zendikar 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 the fifth edition of the D&D rules. D&D is a flexible rules system designed to model any kind of fantasy world. 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 Magic in your role-playing game. The point is to experience the worlds of Magic in a new way, through the lens of the D&D rules. All you really need is 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 Zendikar, 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 Zendikar found on MagicTheGathering.com and the Zendikar plane profile.
The game mechanics in this supplement are usable in your D&D campaign but are not fully tempered by playtests and design iterations. For these reasons, material in this supplement is not legal in D&D Organized Play events.