"[Type]cycling [cost]" means "[Cost], Discard this card: Search your library for a [type] card, reveal it, put it into your hand, then shuffle." This type is usually a subtype (as in "Mountaincycling") but can be any card type, subtype, supertype, or combination thereof (as in "basic landcycling")
Then we cast Zada and the fun begins
Mono-Blue can not ramp up its mana very well and requires extensive use of mana rocks, so Mana Drain adding the mana helps you to start ramping up into your big spells
It forces the opponent to either ante the top card of their library or let the outcome of the game be decided by a coinflip
The plan with this deck is to make sure opponents dont overtake the field, slowly gaining an edge with the value this game creates
For example, the black enchantment Black Market Connections provides a nice source of mana, card draw, and tokens with Changeling that will benefit from the Sliver horde's bonus