Henry VII reigned from 1485 to 1509. He was the first of the Tudor monarchs. His claim to the throne was through his mother. She was descended from King Edward III, who had reigned from 1327 to 1377. This meant that Henry’s claim to the throne was not very strong, but it was based on his family ties.
Henry’s mother was Lady Margaret Beaufort. Her great-grandfather had been John of Gaunt, who was the third surviving son of Edward III. At birth, her grandfather, John Beaufort, had been illegitimate. He was later declared legitimate by Parliament (twice) and by the pope. Her father was also named John Beaufort. He was his father’s second son but he ended up inheriting his titles when his older brother died childless. In this way, Henry could claim to be directly descended from Edward III, though the relationship was somewhat distant and his lineage had originally been illegitimate and barred from the succession.
It is easy to see how Henry VII had a claim to the throne as he was directly descended from a universally acknowledged king of England. What is harder to understand is why someone with such a distant connection to the throne could become king. The answer to that is that Edward III was succeeded by his grandson, Richard II. Richard II was a weak king whose reign led to much conflict between himself and his highest nobles. The conflict led to Richard being deposed by another of Edward’s grandsons, who was crowned Henry IV. He passed the crown to his son, Henry V. Henry V died when his son was an infant. That son, Henry VI, grew up as king but his reign was not stable. There was much infighting over who would influence him in his youth and he suffered from mental problems as an adult, even needing a Council of Regency to govern for him at times. His reign ended in civil war. His only son died in battle at age 17 in 1471, leaving a situation in which the Beaufort family had a stronger claim to the throne than anyone else on the Lancastrian side of what we now know as the Wars of the Roses.
So, Henry VII’s claim to the throne is that he was the great-great-grandson of Edward III. This claim was strong enough to get him to the throne because of the conflict over succession and the lack of clear heirs to the throne.