Short answers, pulled from the story.
Grand Prince of Moscow was the ruler of the Grand Principality of Moscow, a title that evolved from the simpler Prince of Moscow designation used before 1389. The Moscow principality was originally established in the 13th century as an appanage within the Vladimir-Suzdal grand principality.
Daniel of Moscow died in 1303 without ever holding the grand princely title of Vladimir. Under traditional succession practices, this placed his descendants in the category of izgoi, meaning they had no legitimate claim to the Vladimir throne and were outside the recognized line of inheritance.
A yarlik was a formal patent issued by the Mongol khans of the Golden Horde granting the grand princely title of Vladimir. The Moscow princes cultivated Horde favor and received the yarlik by the second quarter of the 14th century, bypassing the dynastic traditions that excluded them from legitimate succession.
The Great Troubles was a decades-long war of succession that engulfed the Golden Horde in the second half of the 14th century. This internal conflict weakened the Horde's ability to manage the Russian principalities and allowed Moscow to grow too powerful to be checked by shifting the yarlik back to the princes of Tver.
In his will of 1389, Dmitry Donskoy transferred the grand principality to his son Vasily I as a hereditary possession, bypassing the khan's traditional right to choose the grand prince and award the yarlik. This act usurped a prerogative the Mongol khans had exercised for generations.
The Golden Horde used Moscow as a counterweight against Tver, which was the strongest principality in the region. Keeping rival powers in competition served Mongol interests, but this strategy backfired when Moscow grew too powerful during the Horde's own internal conflicts.