What happens aerodynamically when we trim an aircraft?
It depends on the aircraft in question, the arrangement of the lifting surfaces, and how they are designed to ensure both static and dynamic stability.

For a simple case, well consider a traditionally designed airplane with a wing set placed approx midway along a fuselage with a horizontal tailplane at the empennage.

The design provides both good static stability and good dynamic stability.

It does so by placing the center of gravity (CG) ahead of the center of lift (CP) for the main wing and causing the tailplane to produce a downwards lifting force to counterbalance the moment caused by the distance between the CG and the CP.  While this is not as efficient a lifting arrangement as other designs like a canard foreplane, it forms a naturally stable platform in terms of longitudinal stability.  The airplane will remain stable only if these two moments cancel each other out causing the Net moment about the CG to be zero AND if the design naturally returns to a zero moment about the CG when dynamically changing speeds and attitudes.

For trimming, here we will only consider pitch trim for longitudinal stability.  An airplane trimmed for cruising at a specific airspeed and a particular altitude will have a net longitudinal moment about the CG of zero.  If the airplane increases its airspeed, the tailplane arrangement is designed to increase its downward force, creating a nose up pitch moment.  This in turn decreases the airspeed causing the the downward tailplane lift to decrease again, resulting in the nose pitching forward again until an equilibrium of forces and moments is again attained, which will be at the original airspeed for which the plane was trimmed for.  To maintain straight and level flight at a higher airspeed the trimming action will decrease the amount of lift that tailplane produces at the new selected airspeed, again resulting in a net longitudinal moment about the CG to zero.

Longitudinal trimming can be accomplished several different ways the first of which is a design feature which can alter the angle of attack of the tailplane in flight to change the lifting force it creates.  This is common on jetliners and other large aircraft but not common on light aircraft due to the excess weight of the structure.  Another option for smaller aircraft is the installation of a movable servo or antiservo tab on the elevator surfaces.  These tabs force the elevators to a new neutral position, again altering the lifting force from the tailplane and creating a trimmed condition.

In regards to the original question the net CP, that is the combined location of both the center of pressure of the wings and the center of pressure of the tailplane will coincide with the CG on the longitudinal axis of a trimmed airplane.  If the airplane accelerates, the NET CP will move fwd of the CG, causing the airplane to nose up, without additional trimming to bring it back.  Similarly, when the airplane decelerates, the NET CP will move aft of CG, causing the aircraft to nose down without additional trimming.