Why did some kings legitimized their children born out of royal wedlock?

They didn't, normally. That was quite rare.

What they often DID do was grant them noble titles and estates. That was not the same as legitimizing them. It gave them social parity with nobles, not royals.

The view was that a king's son or daughter was the equal of any noble but not necessarily the equal of a royal child born in wedlock.

Louis elevated only two of his several illegitimate children to the status of "prince du sang," and he had to push hard to do it. It didn't really matter anyway, since their places in the line of succession followed those of all the legitimate heirs, so they would never have had a chance to inherit the throne.

One of the most interesting cases I know of was that of the children of English King Edward III's son John of Gaunt, Duke of Lancaster, by his mistress and third wife, Katherine Swynford.  When Gaunt was finally able to marry Katherine, their 4 children were legitimated by permission and decree of both the Pope and King Richard II, Gaunt's nephew.  Eventually, a direct descendant of the couple -- Henry Tudor -- became King Henry VII.  The Stuarts, Hanoverians, and the Windsors are all descendants of this "legitimated" line (and also of royal lines that were always legitimate).

But the legitimation would not have happened if Gaunt and Katherine had not married once they were both widowed.

They, the illegitimates, may have been given titles etc., but they were not very often actually made legitimate.   Some might have done this to continue their line.   Most did not.

William the Conqueror (aka William the Bastard) managed to secure his claim to the throne without any great concern about being born out of wedlock. In his case his childless cousin named him as successor, but he had to invade to make good on it. Legitimacy of heirs was seen as a way of avoiding claimants coming out of the woodwork in future generations and avoiding such disputes.

In absolute monarchies, as France was at the time of Louis XIV, a king might choose to legitimise children born out of wedlock, in order to secure succession through their own descendants.  

They may have been given titles but they were not legitimate. Henry Fitzroy, the son that Henry VIII sired with Bessie Blount was made Duke of Richmond and Somerset, but he couldn't inherit the throne. He was brought up as the son of a king, even attending court, his father making no secret of the fact that he could produce a son. This pointed to the failure on the part of his first two wives to oblige him with an heir, until Queen Jane gave birth to Edward VI.

Legitimizing such children was quite a rare thing to do. The king simply wished to do so and there were no laws to prevent the action.

"Legitimize", not "legitimized". Because some kings felt that since their offspring all had royal blood, there was no reason NOT to legitimize them.