There are some modified versions of QM that tries to place this on a more rigorous footing, but none of them have convinced everyone that they do. My personal favorite is the many world's approach that in many ways is simpler than traditional QM because it says that there's no such thing as a measurement. Instead, when you think you're measuring something what you're really doing is entangling yourself with the system you're measuring which means that your state is no longer separate from the state of the system. There's a part of you that sees each outcome.
This is actually already how microscopic systems work: if two particles collide and get entangled, the state of each particle sort of splits in two. The only thing that MWI says is that this dynamics also applies to macroscopic objects.
You measure the colour of an object by bouncing light off it and seeing what comes back. The objects state is modified when the light hits it, since it imparts momentum.
I have to say I am always at a loss when quantum physicists start talking about "measurement".
In the classical world, measuring means looking at a particular variable x in a system S at time t, S(t) and via some process specific to x (which we want to measure), Mx, obtain the value of Mx(S(t)).
In QM by contrast, it seems that measurement itself has an action upon the system so that measuring in fact means looking at some Mx(Z(S,t)) where you actually never know S but only some kind of end product Z that is believed to reflect S but is itself the result of an unknown operation on S that QM people call "collapse".
So you seek Mx(S) but in fact spend your time looking at Mx(Z(S)) and draw conclusions on S... but I have yet to hear anyone explain to me, physically what is Z, how it works, etc. Lots of statistics, but no real understanding of that "collapse" process.