This question is about the interpretation of quantum theory (QT). The most popular interpretation at the moment is Everett's Many Worlds model and this will be used here.
According to QT a particle is present as a wavefunction until it experiences an interaction. A wavefunction can be used to calculate the probability of finding a particle in a particular state (position, momentum etc)**.
Suppose we have a state that can have the values A or B, the wavefunction can be used to calculate the probability of state A or state B at a particular place. According to QT the particle is in state A or B OR in a superposition of A and B (see Qubit ). When the particle experiences an interaction with the environment, such as a measurement, it is measured as being either A or B.
What do we think happens when the measurement is made? According to the Many Worlds interpretation if we measure the particle to have state A this state becomes part of our universe and state B is then lost to another universe.
Quantum wavefunctions can describe bodies of any size. There is even a wavefunction for the universe. Quantum entanglement occurs when a particle becomes a member of a group of particles that exchange information (interact) profusely so that the individual particle can no longer be described by a wavefunction over a meaningful period of time. Instead there is a wavefunction for the whole group. So long as the group is isolated from other groups the wavefunction of the group provides an accurate description of its states.
Suppose we measure a property of an entangled group such as position. At the moment of measurement the group becomes part of our own quantum entangled group and all the other possible positions of the measured group disappear into other universes. If the group we are measuring is large then a whole volume of space is affected instantaneously. We are selecting a particular state of an object from other states. Those states are already there in space as a quantum superposition so we are not shifting an object from one place to another.
In the case of a group of two entangled particles they continue to be described by a single wavefunction until either interacts with another group or the environment (the group of particles of which we are a part). A measurement of one of a pair of entangled photons will immediately select the value of the wavefunction for the other particle that is consistent with this measurement. All the superposed states of the photon group disappear into other universes of particles and we select just one set of states.
Why doesn't it take time for a selection of a particular state of a group to be expressed at a distant point? This is the same question as why doesn't it take time for the rear of a planet to be created after we first observe the front of it. It was always there. After creating an entangled pair of photons, letting them fly off to different places and then making a measurement on one, the complementary state was present at the position of the other photon along with all the other possible states. However, only the complementary state becomes part of our entangled environment once it is selected. The other states are lost to observation. We only observe our own entangled group (our environment) that now contains a particular state of a photon. There are other versions of us for whom this is not the case but we cannot observe them.
That is my understanding of the Many Worlds interpretation, there are other ways of looking at the problem. I am not convinced by any of the interpretations on offer including Many Worlds.
** strictly the wavefunction describes the "amplitude" for a particle to have a state but this is directly related to probability.
PS: Notice that the interpretation is compatible with special relativity. Our own entangled group becomes another group the moment a measurement causes another member to be incorporated. Our entire universe changes and SR applies within this changed universe.