A theoretical use is that it shows that the axioms of the natural numbers (the Peano axioms) aren't enough to pin down what we think the natural numbers should be -- there are these crazy non-standard natural numbers out there!

In context of the discussion, this non-standard model of the natural numbers gives some intuition about what happens if the consistency of ZFC is independent of ZFC and you add in the axiom that ZFC is inconsistent: your natural numbers will have to be something similarly weird.

> you have just made a new type of number not comparable to natural numbers

But they are comparable. The everyday natural numbers are embedded in this new system, which is what these (n,n,n,...) elements represent. One way to interpret what we've done here is to add in infinities to the number system so that all the normal operations still work, and there's even a total ordering on them!

Using the real numbers instead of the naturals, you get the so-called nonstandard real numbers, which is the number system used in nonstandard analysis. Nonstandard analysis is a way to do analysis without epsilon-delta proofs. Nonstandard reals include infinitesimals and infinities, similar to nonstandard naturals having infinities. There are even textbooks on the subject -- I haven't read them, but they claim it makes analysis easier.

The last thing that comes to mind is that nonstandard numbers of this exact type show up when you study certain infinite-dimensional number systems and calculate the points (like in the end of my comment).