How do children understand the concept of number size? How do children understand that one number is larger than the other? For example, given the numbers "8" and "9", how do they learn that "9" is the larger one? This apparently develops as a precursor to addition. Is it memorized like the result of simple addition?
Short answer
It has been argued that infants are born with an ability to recognize,  distinguish and even operate on small numbers.  

Background
Speaking from experience (n = 2 :-) toddlers learn to count, first of all. That means they know 2 comes after 1, 3 after 2 etc. In conjunction, we read them stories, like the very hungry caterpillar that graphically shows how, consecutively, the little critter starts eating holes in 1 apple, 2 pears, 3 plums and so forth, after which the caterpillar turns into a butterfly (Fig. 1). So there's the element of a growing and developing critter (time) eating more and more pieces of fruit (size). Meanwhile the parent will count the ever growing number of fruit out loud. These kinds of books are highly appealing to 2-year olds in my experience. My 3-year old can count, and has a reasonable idea of what it means. My 5-year old can perform simple mathematical operations, and will have a reasonable idea that 2+2 is less than 4+4. 

So much for the anecdotal evidence - In the literature it has been argued that infants are born with an ability to recognize and distinguish among ones, twos, and threes, and can even operate on very small numbers, e.g., recognize that one object added to another makes two - all before they develop verbal-based counting competencies. It may in fact go so far that infants are innately born with counting principles, allowing them to count nonverbally (Baroody et al., 2005)

This information leads me to believe that all the effort (us) parents put into reading the 'Very Hungry Caterpillar' is more to associate language to mathematics, i.e., to provide the lexicon to talk about math, rather than to educate them on maths per se. 


Fig.1. Graphical content of 'The Very Hungry Caterpillar'. source: Little Folk Visuals 

Reference
- The development of young children's early number and operation sense and its implications for early childhood education. In B. Spodek &amp; O. N. Saracho (Eds.), Handbook of research on the education of young children (187-221). Mahwah, NJ, US: Lawrence Erlbaum Associates Publishers