The measurement of an operator A on the state |ψ⟩=(|0⟩+|1⟩)/2–√, either gives the eigenvalue a0 associated with the state |0⟩ or a1 associated with |1⟩. Isn't this natural? How does it make sense to get a superposition after a single measurement? What on earth does that mean? What would the result of a single measurement be if superposition were retained?
Yes, this is natural, and it is not a problem. The quantum state is simply an expression of the possible results of measurement. After the measurement a definite result is known, so the state is no longer a superposition.
The problem arises in interpretations which ascribe some kind of physical reality to the wave function (state). This would require an instantaneous transition of the physical state, which is not only unexplained but also violates fundamental principles in relativity. The problem is resolved if one thinks of the state as being simply a statement of the possibilities for the result of a measurement, not as a description of physical reality. But then one has a different problem, namely to explain why the Schrodinger equation is obeyed.
This can be explained with mathematics, but it does not take just a little bit of mathematics, unfortunately. The problem is actually not with measurement, but with explaining why the Schrodinger equation is obeyed for a system which is not governed by underlying determinacy (hidden variables).
The starting point is the principle that measurements have probabilistic outcomes. This is actually true even in classical mechanics (as described in standard error analysis). One can then set up a general probability theory for measurement outcomes. One defines states to describe possible measurement results, and applies natural structures of language to establish a Hilbert space, and relates the inner product to probability to obey the Born rule.
One then establishes that to maintain the probability interpretation under time evolution, unitarity is required, and that the conditions for Stone's theorem are obeyed, from which the Schrodinger equation follows as a simple corollary.
I have given a complete treatment in The Hilbert space of conditional clauses