Because if I hire that person, they'll be in charge of critical production systems and will face much harder problems in much more stressful situations and if they can't handle the palindrome question, I have very little confidence they can handle the actual job.
Not claiming to be that hero, but I have personally had those moments where I came out of the interview, and as the disorienting fog of pressure lifted during my drive home I said to myself, "WTF happened to my brain in there? I know that stuff cold!"
EDIT: I think I read this one from a commenter here, but it's so true: We're interviewing people to be music composers, but judging them by their ability to be performance artists.
(i.e. bring out groovy, using bash, etc)
Is it reasonable to ask things like that? Maybe, maybe not.
But as stressful as an interview is, it's still a basic problem (I can English describe it to you in 30 seconds...coding (in C/C++) would take me another 5-8 minutes (and, probably, be pretty damn inefficient ... but it's still a simple problem)
Throughout my career, I've been in professional situations where I was under a lot of pressure: Having to explain failures to executives, near-impossibly tight deadlines, production outages. At least I haven't had to testify in front of Congress yet. Absolutely NOTHING I've encountered in my professional day-to-day has the extreme level of pressure of a typical silicon valley job interview.