To add to ArtforLife's answer you are speaking of the famous Quantum Measurement Problem. I disagree with his/her answer a little (even though I upvoted it) insofar that we're not sweeping the question under the rug so much as using abstraction to decouple descriptions from one another so that we can tackle and conceive of one description at a time. In other words, Physics is very aware of the depth and importance of the question and it is an open and very active problem, but one can make considerable headway in theoretical and experimental physics without answering it through the appropriate abstraction. Namely, we postulate the mathematical idea of an observable to model this measurement process, which is an Hermitian operator together with a recipe for how this operator corresponds to the measurement process: the quantum state on measurement somehow winds up in one of the operator's eigenstates straight after the measurement; which one is random but the probability distribution describing the statistics of which eigenstate arises from the measurement is calculated through Born's probability interpretation. Notice how through this abstraction we decouple our outcome statistics calculation from the measurement problem. It is a simple "black box" kind of model, and the question of whether it is owing to anything as complicated (and as poorly understood) as "consciousness" is put aside for the calculation.
As to what goes on at "measurement", there are several current theories people are thinking about. There are a couple of posts on this site about how entanglement of the measured process with the quantum state of the measurement apparatus can give effects that are very much like the "discontinuous" state change that an observable-modelled measurement begets. See:
DanielSank's answer to the question "Really how can an observable quantity be equal to an operator?"
Lurscher's answer to the question "Is quantum entanglement functionally equivalent to a measurement?"
These ideas are further formalized and studied in the theories of Einselection and Quantum Darwinism. Essentially all these ideas fall under the general explanation that measurement may simply be a loss of knowledge that goes with the interaction between a known quantum state and an incompletely characterized, complicated measurement system.
I give the ideas above not so much to imply that the "measurement problem" is on the brink of being solved, but rather that (1) a solution is indeed plausible within even my lifetime and (2) that the problem is given ongoing serious thought rather than "swept under the rug".