Why do pilots say this is my aircraft?

A pilot will say "my aircraft" when he or she has taken authority of the flight controls - usually but not necessarily from another pilot in the cockpit who had previously been in control. It is a standardized verbal message, and a concise way of saying, "I am now steering the aircraft and you are not."

It is particularly useful if one pilot feels he or she must aggressively take control of the aircraft from the other pilot in order to prevent impending harm to the aircraft or its passengers. For instance, most airlines require the pilot-not-flying to call-out "80 knots" during takeoff when the aircraft has reached an airspeed of 80 knots rolling down the runway. They also require the other pilot, the pilot-in-command, to respond with "check" as a way to confirm that the pilot-in-command is absolutely focused on taking off and putting the aircraft into the air, one of the most dangerous segments of flight. (Sometimes, as many pilots can attest, one may be physically capable of pushing the throttle forward and keeping the aircraft centered along the runway, but one's mind is not entirely "there," which can be very dangerous.) If the pilot-in-command does not respond "check," the pilot-not-flying will say "my aircraft," take control of the flight controls, and, depending on airline policy and runway conditions, either takeoff normally, or execute an aborted takeoff.