Heisenberg describes the processes involved in measurement as follows:
The observation itself changes the probability function [wave function] discontinuously; it selects of all possible events the actual one that has taken place. […] The transition from the “possible” to the “actual” therefore takes place during the act of observation. If we want to describe what happens in an atomic event, we have to realize that the word “happens” can apply only to the observation, not to the state of affairs between two observations.
(Werner Heisenberg, Physik und Philosophie, Ullstein Frankfurt (1959), p. 37)
If you read on at this point, however, you have to exercise caution in one respect. Heisenberg’s conception of the wave function does not correspond to the one that is widely accepted today. According to Heisenberg, the wave function represents our knowledge of the quantum system. Through observation, our knowledge of the system changes discontinuously.
Today, most physicists no longer hold the view that the wave function represents our knowledge (this statement is of course true in the same trivial sense as the statement that, in classical mechanics, specifying the position coordinates as a function of time represents our knowledge of a thrown stone).
Rather, it is held that the change in the wave function during the measurement process corresponds to a real change in nature. For example, the presence or absence of interference fringes should not depend on our subjective knowledge of the quantum system.