Interesting question!

It seems quite likely to me that two leading factors are that while they were at one point "the bastard descendants of a third son", they were a) legitimized, though still prohibited from being in line to the throne, and b) the cousins of Richard II and the half-siblings of Henry IV. In regards to point A - there's a big difference between various half-siblings a royal duke has with a number of women, a closely-related set of children he has with his one mistress, and the children he has with his mistress, whom he then marries, and whose children he has declared legitimate, as though he'd been married when they were born. By 1397, when they were legitimized, they were simply royal cousins almost unlike any other.

Now for point B, though. They became royal siblings when their fully legitimate half-brother, Henry Bolingbroke, usurped the throne from his nephew, Richard II, and became Henry IV. This put them in a rather extraordinary position: not only were they close to the throne, they were barred from inheriting it, which essentially made them the most trustworthy advisors and assistants a king could have in a period where the succession was wildly unstable. And despite their lack of ability to inherit the throne, that level of access to the king gave them power and standing.

It would later go on to be a huge part of Henry VII's claim to the throne. He was descended form the Beauforts on his mother's side, which was not itself a hugely useful situation as the family was barred from the throne, but as I discussed in my answer to Why did medieval kings go through so much trouble to establish that their right to rule was legitimate? ...

Henry VI's mother, Catherine of Valois, was Henry VII's grandmother. That did not, in and of itself, give him a claim to the throne, but it put him firmly into the orbit of the royal family and therefore the inner circle of the Lancastrian cause. In addition, he had royal blood on his mother's side as well - she was descended from (Prince) John of Gaunt, through one of his legitimized illegitimate children. However, this was problematic in and of itself.

...

I argued in the previous answer that Henry VII's Beaufort ancestry was not the single deciding factor in his being perceived as the heir to the Lancastrian cause, and I still stand by that, but it was clearly a factor. If Henry VII's father hadn't been a Tudor half-brother to the king, he would probably not have had enough of a spotlight or been closely connected with enough powerful people to win the throne. On the other hand, without his mother's status as the only child of the eldest (surviving) son of the eldest Beaufort son, it's unlikely that he would have been able to present himself as a real contender, despite the legal bars to Beaufort inheritance.

Simply being related to the king and allowed to be emotionally near him was a very big deal. As a result, it's unsurprising that the Beauforts were highly influential and present in political events of their time.