The question should really be, why did Armstrong pilot the Lunar Module (LM) manually, and the autopilot was switched off. According to Tales From The Lunar Module Guidance Computer by Don Eyles, Armstrong and Aldrin never switched places during the LM powered descent and landing phase. Armstrong was on the left side of the LM cockpit where the manual controls were, and Aldrin on the right and responsible for working the DSKY (Lunar Module Display and Keyboard Unit), essentially instructing the flight computer.
For reference, here's a photo of the Apollo Lunar Module's interior (source and credit: NASA HQ):
It seems that the 1202 Program Alarm that the LM was experiencing prior to the landing prompted the switch from
ATT HOLD flight mode. From mentioned source:
ATT HOLD meant the digital autopilot's Rate-Command Attitude-Hold
mode, in which the astronaut could command an attitude rate by
deflecting a joystick. After the stick was released the autopilot
nulled rates to maintain the present attitude.
Armstrong, as the mission commander, had to make a decision whether to proceed with the landing or not, while dealing with it with the mission control in Houston. It turned out that:
Due to an error in the checklist manual, the rendezvous radar switch
was placed in the wrong position. This caused it to send erroneous
signals to the computer. The result was that the computer was being
asked to perform all of its normal functions for landing while
receiving an extra load of spurious data which used up 15% of its
time. The computer (or rather the software in it) was smart enough to
recognize that it was being asked to perform more tasks than it should
be performing. It then sent out an alarm, which meant to the
astronaut, I'm overloaded with more tasks than I should be doing at
this time and I'm going to keep only the more important tasks; i.e.,
the ones needed for landing...
— Margaret Hamilton, lead Apollo flight software designer
And the computer was prompting Armstrong and Aldrin with other alarms, for all of which they received a go to proceed from Houston. According to Human and Machine in Spaceﬂight, David A. Mindell, 2008 Massachusetts Institute of Technology (PDF):
With the push of a button, the astronaut could abort the landing, an
action practiced countless times in simulation. Yet he held off.
Armstrong later explained himself as a mechanism:
‘‘In simulations we have a large number of failures and we are
usually spring-loaded to the abort position. And in this case in the
real ﬂight, we are spring-loaded to the land position.’’
Houston checked out the problem. Young engineers recognized it from a
recent simulation, and conferred with their support teams in the back
room. They quickly found the cause. The computer was overloading and
restarting but not shutting down. It was ignoring low-priority tasks,
but these were not critical for the mission. ‘‘We're go on that
alarm,’’ the ground controller replied, meaning the LM could proceed.
For his role in clearing the landing, engineer Steven Bales later
accepted a presidential award on behalf of the ﬂight control team.
Armstrong surveyed the computer display; it had frozen. He checked the
LM’s systems. The vehicle seemed to be responding to his commands,
meaning the computer was still running. So he continued.
Emphasis mine. So it seems that the switch to manual was due to a set of circumstances prior to the landing. Armstrong, as the mission commander, continued piloting the LM in
ATT HOLD flight mode after the 1202 Program Alarm, and all subsequent alarms they were in contact with Houston about. After mission control gave a go on the alarms, and having the control over the flight computer, he made a command decision and took The Eagle down to its landing spot — Tranquility Base at the Sea of Tranquility (Mare Tranquillitatis), the Moon.