Quests have historically been insanely popular in Hearthstone history. Even the weakest of quests tend to see play and a surprising amount of it because people like the idea of working toward something specific in the game. It scratches much of the same itch as building a Combo deck but gives you a lot of structure for where to start and what direction to take your deck in.
Historically though, Quests have been relatively narrow and put a lot of deckbuilding restrictions on the player. If you’re going for the Making Mummies Paladin Quest in current Hearthstone, you’re going to be jamming your deck full of Reborn minions and stuff you want to duplicate; otherwise, there’s no point, and you’re wasting your time.
Side Quests turns this paradigm on its head a bit, with the goal being to reward the player for playing into their class’ core identity while being more accessible and working closer to how normal cards work. Each still costs 1 mana (at least the so far revealed ones do), but they do not start in your hand and must be drawn usually. Of note, the Side Quests are not Legendary, so you can run multiple copies (and this likely also means you can Discover them with some effects).
Each has relatively gentle accessibility requirements. The Mage Sidequest, Learn Draconic, summons a 6/6 dragon if you spend 8 mana on spells, while the Hunter Sidequest, Toxic Reinforcements, summons three 1/1 Leper Gnomes after you use your Hero Power three times.
This should be hopefully revitalized certain partially defunct playstyles, with Hunters being encouraged to SMOrc more and Mages encouraged to spend mana on spells (instead of generating infinite value with 0 mana spells). Unfortunately, the currently revealed ones feel a bit weak. Learn Draconic is a 9 mana 6/6, while Toxic Reinforcements certainly feels like a “win more” card. If you have the luxury of hitting your Hero Power three times, you might be ahead enough that the reward is mostly pointless.