This is the deck that I have been using to run rings around the Druid and Paladin decks that are currently flooding the meta.
It is a combo deck that looks to pull off the usual Unlimited Fireballs.
As such, the combo pieces you are looking for to do this are Archmage Antonidas, the Sorceror's Apprentices, and the Potion of Illusion.
To get to the stage where you can pull off the combo, you need to use the Tortollan Pilgrims to repeatedly pull duplicates of itself (with a Potion) so you have 1-cost versions in hand. You can then multiply these up along with Sorceror's Apprentices to be able to stall with "endless" Blizzards and Frost Novas on the opponent's board and build the Exodia pieces.
To achieve your goal, the early game minions are there to provide a way of stalling and controlling the board until you can get the bits you need.
If you ever find yourself having pulled your spells out by mistake into your hand, use the Eleks to shuffle more copies into your deck.
It's a little bit of a complicated win, but it is usually pretty consistent if you can stall the early game. By all means ask me to explain anything you need.