In the Bug Fixes section all the way at the bottom:
[Quests] Some Daily and Weekly Quests will now include an arrow at the bottom. Press this arrow to be taken directly to the part of the client where that Quest can be completed.
It means that clicking it will take you to the relevant game mode. I had a quest to play Tavern Brawl and when I clicked it, it took me to Tavern Brawl. The weekly arrow will bring you to Ranked mode (not sure which though, I assume Standard).
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