Early in the 15th century, the Tudors apparently were staunch enemies of the Lancasters. How did they come to support and even perpeatuate the Lancastrian cause in the War of the Roses?

I assume you’re referring to the 1400-1415 uprising of Owain Glendower? Meredudd (Meredith) ap Tudur and his brothers were cousins to Owain and joined his revolt, but the whole endeavor was aimed at Welsh emancipation from English rule, Lancastrian or otherwise. Edward I had pretty thoroughly conquered Wales in the 13th century; Glendower’s uprising was Wales’ last bid for independence. Political instability following Henry IV’s (Lancastrian) usurpation of Richard II provided the impetus for revolt, but in Wales the sentiment wasn’t specifically anti-Lancastrian as the term would come to mean later in the 15th century. At any rate, the revolt failed, Meredudd ap Tudur was deprived of his lands, and Wales went back to being a quietly bitter, dejectedly loyal subject of the English crown.

By the time ‘Lancastrian’ and ‘Yorkist’ became fighting words, the family of Meredudd’s son Owen was very much in the Lancastrian camp. Technically, his name should be 'Owain ap Meredudd ap Tudur' based on Welsh patronymics, and the Tudors should probably have been called the Merediths, but in an England of varied spellings and emerging surnames, he ended up as Owen Tudor. Not much is known about his early life. He was born during the early part of Glendower’s uprising. After his father’s downfall he was pardoned by Henry V and fought in the king’s service at Agincourt.

Owen was tall, handsome, charming, and many other fine adjectives as well. Shortly after Henry V’s untimely death, Owen became the lover of the king’s young widow, Queen Catherine of Valois, mother to the infant Henry VI. They were married in secret, to the great consternation of the English nobility (Catherine was, to put it mildly, way out of Owen's league). They had at least two sons, Edmund and Jasper Tudor, half-brothers to the king. Catherine died young. In the absence of her protection Owen fell on rough times. He was imprisoned, escaped, spent some time on the run, was re-captured and again imprisoned. Henry VI finally pardoned him. Edmund and Jasper were officially recognized as half-brothers to the king and both became Earls, very much politically beholden to the the Lancastrian power.

Edmund Tudor married Margaret Beaufort, a descendant of John of Gaunt (Duke of Lancaster, son of Edward III), and after the aristocratic bloodletting of the Wars of the Roses, their son Henry Tudor was the last of the Lancastrian claimants to the throne, however tenuous that claim certainly was.