Bit of a complicated answer given that I don't think you mean Henry VI in particular.

Firstly, the situation revolves around the royal house hold and in particular Henry VI and his mother. His mother was a widower and at the time not constrained in future marriages (legally). She was suspected of beginning a dalliance with Edmund Beaufort, Duke of Somerset. Their potential marriage would have skewed the internal politics of the realm greatly as Somerset was already one of the more powerful lords in the realm and having access to the King through his mother was something frowned upon by other lords. They quickly passed a law that forbid Queens from being married to landholders in England/English landholders. She was pregnant, some suspect of Beaufort and quickly married Owen Tudor. Tudor is a special case, because he was Welsh. The Welsh at the time were looked down upon, indeed, Tudor would later need to be made into an "honorary Englishman" by Parliament.

The second focus is Henry VI as a personality. He was a relatively kind king. Mainly through his own incompetence. It's doubtful he even knew of the existence of his half-brothers until his mother's death. After which they were brought to court. Both Owen and his children were granted gifts and lands at different points by Henry VI, who essentially had no other immediate family. They'd prove their worth through the beginning of the War of the Roses and remain loyal to the Lancastrian cause. Owen would in fact be executed after a lost battle against the Yorkists.

The fortunes of the Tudors rose and fell with the Lancastrian cause overall. When the Yorkists came to power for a longer stretch after 1471 the Tudors fled to the continent. They were used as pawns in a power game between Edward IV of England, Louis XI of France, and Duke Francis of Brittany.

When Edward IV died in 1483, his young son inherited the throne. His brother, the soon to be Richard III, was made Lord-Protector as he had proven his loyalty and worth during the Yorkist struggle for the throne during the War of the Roses. Richard would soon usurp the throne and the young king and his brother disappeared soon after. The disappearance of the children as well as the death of his wife cast much suspicion on Richard III. His political capital was further eroded by the rumor he intended to marry his niece (the future wife of Henry VII).

The Tudors would return to English soil in 1485 and play upon Richard's unpopularity. Even so, Richard's army still outnumbered the Tudor force. The battle of Bosworth was decided mainly due to the fact that Richard III was separated from the main body of his force and killed. Combined with Lord Stanley's betrayal this sealed the fate of the Yorkist cause and allowed the Tudors to come to power.