What is the meaning of 么? If you want to say "This dog," you'd say, "这只狗". If you want to say "This thing," you'd say, "这个东西". If you want to say, "This much," you'd say, "这么多".

I understand the reasons for the 只 and the 个, but what does the 么 really mean?
么 has lost most of its lexical meaning in modern usage, but it appears to have originally been derived from the word 物, meaning 'thing'. Chinese by Jerry Norman, pp. 119-20 makes mention of the etymology of various modern words featuring 么:


  In an insightful article, Zhāng Hùiyīng (1982) has shown that shénme 什么, the standard modern word for 'what', comes from shí wù 什物 (MC źjəp mjuət). This form (in a number of different phonological shapes) makes its first appearance in the Tang dynasty. Shí wù 什物 originally meant 'vessels, household utensils, things (of various sorts)'. The development of a interrogative meaning 'what' from a meaning 'thing' finds a parallel in modern Italian, where cosa 'thing' is also used to mean 'what'. ...
  
  ...
  
  [Regarding 'how'...] In the late Tang and Wudai periods, a different form begins to be found; it consists of the verb zuò 作 (MC tsuo-) plus one of several variants of the morpheme meaning 'thing, what' discussed in relation to the origins of modern shénme 什么 ... In the Wudai period, a fusion word zěn 怎 (from an earlier *zěm) based on these forms appears for the first time. It is this fusion form that is directly ancestral to the modern zěnme 怎么 'how'. Zhème 这么 'in this way' and nàme 那么 'in that way' were probably analogically based on this form.


Notes:


The article by Zhāng Hùiyīng 张惠英 referenced in that excerpt refers to an article entitled 释「什么」in《中国语文》1982年第4期, pp. 302-5. I wasn't easily able to obtain a copy of that article though.
MC in that excerpt refers to how the words were pronounced in Middle Chinese (ca. 600-1000 AD).


While it's interesting that 么 in these words was derived from a word meaning 'thing', it's essentially a bound morpheme that is no longer productive and can't be used on its own. In effect, its modern usage doesn't have any particular meaning in practice. In many ways, it could be considered a cranberry morpheme.