The simple answer is because Henry Tudor (Henry VII, first Tudor King) was descended from the Plantagents via the maternal line, so hadn't inherited their name (and in fact, surnames were rather more nebulous at this point anyway, but that's rather more complex).

This is the same way that our current monarchs were (until they changed their name) the Saxe-Coburgs, despite Queen Victoria being from the Hanover dynasty. As she married Albert, her children had the surname Saxe-Coburg.

Unusually, Churchill asked Elizabeth II to keep her name (Windsor) and for their son to be Charles Windsor, rather than Charles Mountbatten. Under normal tradition, Elizabeth should have taken her husband's name, and become Elizabeth Mountbatten, and we'd be ruled by the Mountbatten Dynasty.

Back to Henry Tudor.

His father was Edmund Tudor, son of Henry Tudor (a page) and Catherine of Valois (widow of Henry V of England, and daughter of the King of France). As Edmund Tudor was the younger half-brother of Henry VI, he was made Earl of Richmund by Parliament and declared legitimate, as Catherine and Henry Tudor declared that they had definitely not been sleeping together out of wedlock, but had secretly married.

So Henry Tudor was the (half-)nephew of Henry VI, but that's not the main reason he claimed as his right to the throne. It was instead through his mother, who was grandaughter of John Beaufort. John was the bastard son of John of Gaunt (though later legitimised... though later it was declared that while legitimate, his descendants couldn't inherit). His surname was probably given to him because John of Gaunt was lord of Beaufort. John of Gaunt was a Plantagenet, son of King Edward III.

So, what we have here is a Plantagenet, who has a bastard son. His bastard is a Beaufort, not a Plantaganet, though later declared legitimate. Even if their family had been referred to as Plantagenets, when his granddaughter Margaret married Owen Tudor, she took his name Tudor. Their son was therefore Henry Tudor.

You may notice of course, that Henry Tudor was descended on his father's side from the King of France... but not the King of England, only the King's wife. On his mother's side, he was descended from a legitimised bastard (whose family was banned from inheriting the throne) of a Plantagenet. His claim, therefore, was pretty weak. However, by this time, there were no better Lancastrian candidates as they were all dead. Once he'd defeated and killed Richard III, he was a possible candidate who'd just beaten the King and had a big army, and so Parliament declared him King. He married Elizabeth of York, daughter of Edward IV and one of the only remaining Yorkist candidates for the throne, to ensure there was no great opposition. This meant that Henry VIII (and his other children) were descended (if not patrilinially) from both the York and Lancastrian branches.