For that we have to go back to Edward the III.

He had 5 sons survive into adulthood. The throne first passed through Edward III’s oldest son Edward the Black Prince. The Prince had died, so it passed on to his son Richard II. We’ll call this Line 1 and we’ll leave it there for the moment.

Edward III’s second son, Lionel of Antwerp Duke of Clarence only had a single daughter, Philippa, 5th Countess of Ulster. We’ll call this Line 2 and we’ll come back to her in a moment.

The next two of Edward III’s sons are John of Gaunt the Duke of Lancaster and Edmund of Langley the Duke of York. These two men established the Houses of Lancaster and York who were the primary “sides” in the War of the Roses. We’ll call those Lines 3 and 4.

So Richard II has a controversial reign and is overthrown by his cousin Henry Bolingbroke (John of Gaunt’s son). This ends the legitimacy of Edward III’s oldest son’s line. And the throne passes to Line 3 the Lancasters.

The problem is that Line 2 still exists, and in fact exists in a member of Line 4. One of Edmund Langley’s sons had married his cousin Philippa. Their son Richard of York thus claimed that as the heir of Lionel he had primacy over any descendant of John of Gaunt. That sets up the classic Lancaster vs York fight.

However on top of all of this Edward III’s third son John of Gaunt Duke of Lancaster had a mistress, Katherine Swynford, back in the day. After the death of his legitimate wife he married his mistress and had Richard II and the Pope declare his kids through Kathrine legitimate after the fact. The first of these John Beaufort was made Earl of Somerset. And although he is explicitly excluded from succession, the Beauforts are accepted as a cadet family to the House of Lancaster.

Then the War of the Roses happens, and the vast majority of Edward III’s descendants die, one way or the other. However there’s still some Beaufort’s around. In particular John’s son John has had a daughter Margaret Beaufort, who had married several men (not at the same time) including Edmund Tudor, Earl of Richmond. Her son Henry Tudor was the man who become Henry VII.

So to sum up: Edward the III had several sons including John of Gaunt who founded both the House of Lancaster and the House of Beaufort. His son John Beaufort also had a son John Beaufort. That John had a daughter Margret, who gave birth to Henry Tudor. Or Edward III -> John -> John -> John -> Margaret -> Henry.

Henry was pretty much just the last great-great-great-grandson of Edward III left standing.