I'm no expert, but I can think of two general reasons: risk of incapacitation, and general physical limitations (reaction speed, stamina, the ability to maintain concentration for extended periods etc.). The age 65 limit only applies to part 121 ATP pilots, by the way; you can still be a commercial or private pilot for as long as you can keep getting a valid medical.
Older people are simply at greater risk of heart attacks and many other medical conditions, and they're also much more likely to require medication that may not be approved by the FAA or other national authorities. Of course that doesn't mean that every older pilot is the same but I guess that the FAA has decided that since the risks increase with age, at some point they become unacceptable. To understand the specific reasons for that decision you would probably need to dig up the original NPRM or transcripts of the public debate/hearings, but I have no idea if that's possible or not.
Then the other aspect is general physical and mental condition. They both deteriorate with age, and the NTSB sometimes mentions that as a potential factor in accidents:
On the day of the accident, the pilot had flown several formation
sorties. The day was hot, and the formation activities were demanding.
The accident flight was relatively long, and the co-owner of the
airplane guessed it was about 45 minutes long. Most formation sorties
were 30 minutes or less. The accident pilot was in his early 70s and
was by far the oldest member of the flight. Due to the pilot’s age,
the demanding schedule, and the length of the formation sorties,
fatigue could have been a factor, although it could not be
Of course, any discussion like this is a generalization and I'm sure it's easy to find individual 70-year old pilots or controllers who are 'sharper' than younger ones (after all, experience counts for a lot too). But there's no escaping the overall pattern that older people in general are more prone to medical issues and less physically/mentally capable than younger ones, and at least one US court has ruled that there's no effective way to determine individual risk so a general rule is not discriminatory:
The court also concluded that ExxonMobil had established, through the
testimony of several medical experts, that the risk of sudden
incapacitation in flight significantly increased with age, and that
there are no adequate means of individually testing each pilot to
determine whether a pilot was at risk to suffer such an