As a follow-up to today's discussion of HOAS, here is another example: https://en.wikipedia.org/wiki/Higher-order_abstract_syntax#Use_in_logical_frameworks
That example is based on Twelf, which for the purpose of that example, you can think of MMT. In fact, Twelf can be seen as a predecessor of MMT.
Apparently, there's also this classic paper by Pfenning and Elliott on HOAS. The paper's presentation might be a bit too far away from MMT for it to be easily understandable to you. But Section 2 ("motivating examples") might still be of interest to you: it mentions problems of substitution and variable occurrences, which you usually have if you're representing bound variables as mere "string references". I think you might be able to understand bits even if not all of the syntax. (Neither do I understand their syntax by mere glancing.)
By employing, HOAS we completely forego those problems by using meta-level functions (namely from LF). (Internally, these do work with "string references", though, cf. OMV in MMT source code.)