According to MWG Proposition 6.B.2, it illustrates that the expected utility form is preserved only by increasing linear transformation.
What is the significance of this proposition?
The part I find challenging in connecting the dots is when right after the proof of this proposition, the authors claim that this proposition allows us to interpret meaning in differences of utilities. How are these two exactly connected?
This is a subtle topic. Any strictly increasing transformation will preserve the ordering of lotteries. This is the standard result saying that utility is just an ordinal thing, we really don't care about the level of it.
In the other hand, there are representations os someone's preferences that are easier to work with, from the calculations point of view, or improve our understanding of the underlying economics of the problem. One example is when we have quasilinear preferences on good 1 (check MWG, chapter 3 for this example). When this is the case, we have that $U(x_{1},...,x_{I}) = x_{1} + \phi(x_{2},...,x_{I})$. Of course, if we take, say, the exponential of $U$, we will preserve ordering, but not this very useful representation.
When it comes to chapter 6 and lotteries come on the stage, the same happens. One's utility over a lottery (think about it as a distribution of consumption over different scenarios) might be a very complicated function. Maybe, the utility you get by holding an umbrella if it starts to rain also depends on the utility you get when your football team wins and you are not at home watching the game. The V.N.M. utility representation is useful exactly because it separates utilities of different scenarios.
So, when you start comparing lotteries, the effect of an increase in the probability of holding an umbrella when it rains in a given lottery is completely isolated from the effect of a higher probability of another scenario. If we keep consumption fixed over scenarios and let only the distribution of probabilities describing what can happen vary, the V.N.M utility form will let us isolate effects of different probabilities over scenarios and express difference of total utility by difference in probabilities over scenarios between two in-comparison lotteries.
The reason behind only linear transformations conserve this format is that the V.N.M utility form is linear, by definition, and non-linear transformations of linear functions are not linear. This is somewhat related to Jensen's inequality, from my perspective (that's at least how I intuitively interpret that).
I hope this helps!
