What does it mean to have a strictly increasing transformation in consumer theory? We're discussing utility functions
If I'm understanding your question correctly, you are referring to increasing transformations of a utility function. Suppose I have a set of alternatives $X$, a rational preference relation $\succsim$ on $X$, and a function $u:X \rightarrow \mathbb{R}$ which represents this preference relation. Then it can be shown that for any $v:\mathbb{R} \rightarrow \mathbb{R}$ which is monotonically increasing, $v \circ u:X \rightarrow \mathbb{R}$ also represents these preferences. Try to prove this; it's a good exercise. 

So in classical economic theory, we never have a "unique" utility function which represents preferences. This is because we can always take an increasing transformation of the function and get something else which contains all the relevant information with respect to our preferences.