Note that our
language L (see
NSA: what's a language? for more details of
that) only has
names for M, not for *M. Thus, any
sentence S of the language has an interpretation in both *M and M; we're guaranteed that it will receive the same
truth value in both.
In particular, while speaking in L, the language of M, we can only ever access elements of *M in a sentence by using quantifiers. So while we can deduce (and it is true) that there exists an element of *R (the pseudo- real numbers) which is positive and smaller than any positive real number, this element has no name. Worse, we cannot even express this property ("there exists a positive x smaller than any positive real number") in our language L. The transfer principle shows this: if sentence S expressed this property, it would be true in *R, but false in R (there is definitely no such positive real number; half of it would be smaller!), contradicting the transfer principle.
We could conceivably extend L to the language *L, which will have names for all objects of *M. A sentence T in the language *L has an interpretation in *M, but if it contains any constants not in L then it is meaningless in M.