Since the Commission on Presidential Debates is a private organization, the ultimate answer to this is: Because that's how they want it to be.
15% is a rather arbitrary figure, and if someone was polling at 14.7% I would agree that it would be against the spirit of free and open elections to deny them a chance to air their views on such a large stage during a pivotal moment of an election. Still, since this is America and anyone can get it in their mind that they want to be President, there has to be something to use as a delimeter and publicly available polling data from reputable sources is just as good as anything else.
As far as why a private 501(C)(3) organization gets to make the determination that 15% is the threshold a candidate must meet in order to receive a debate invitation, I think the reason is because they actually spoke up and did the hard work of organizing the debate. It is their show that they put on for our benefit. I read a bit of history on the CPD's About Page, where they state this:
After the Kennedy-Nixon debates in 1960, there were no such debates in 1964, 1968 and 1972. There were debates in 1976, 1980 and 1984, but they were hastily arranged after negotiations between the candidates that left many uncertain whether there would be any debates at all. The 1984 experience, in particular, reinforced a mounting concern that, in any given election, voters could be deprived of the opportunity to observe a debate among the leading candidates for President.
In response to the Harvard and Georgetown studies, the then-chairmen of the Democratic and Republican National Committees, Paul G. Kirk, Jr., and Frank J. Fahrenkopf, Jr., respectively, jointly supported creation of the independent CPD. The CPD was incorporated in the District of Columbia on February 19, 1987, as a private, not-for-profit corporation to "organize, manage, produce, publicize and support debates for the candidates for President of the United States."
So why must we listen to the Commission and follow their rules, seemingly arbitrary or not? Because no one else was doing it, and they were able to create a consensus that it needed to be done and at the same time got the two major political parties to agree to let them out of a desire to better inform and educate the electorate without the uncertainty of having to rely on the candidates themselves to work out the details.
Imagine if Clinton and Trump had to agree on the format of even a single debate. I don't think they would, and we would not be able to enjoy the spectacle, or become better informed citizens, of course.