Development and function of spindle neurons
The spindle-shaped brain cells you refer to are called von Economo neurons —named after the man who first described them  (source: Smithsonian). 


In humans, von Economo neurons reside only in the anterior cingulate cortex (ACC) and the frontal insula (FI) (Allman et al., 2011). These two regions are particularly active when people experience emotion and they are reportedly important for "self-monitoring," such as noticing bodily sensations of pain and hunger, or recognizing that one has made a mistake. The ACC seems broadly involved in nearly every mental or physical effort (Smithsonian). By contrast, the FI may play a more specific role in generating social emotions such as empathy, social awareness, and self-control (Allman et al., 2011), but also trust, guilt, embarrassment, love, and even the sense of humor Smithsonian). 
Humans and the great apes (chimps, bonobos, gorillas and orangutans) have von Economo cells. Lesser primates, such as macaques, lemurs and tarsiers, do not. That means the neurons evolved in a common ancestor of all the great apes about 13 million years ago, after they diverged from other primates but well before the human and chimp lineages diverged about six million years ago (Smithsonian). However, elephants and whales have them too, possibly explaining their complex social behavior (Allman et al., 2011). 
I couldn't find literature on the development of these cells in children. Note, however, that, as far as I know, the presence of these cells can be only detected using microscopic techniques, i.e., on coupes of post mortem tissue (e.g., Santos et al., 2010)). This will make research in infants and children difficult.


References
- Allman et al., Ann N Y Acad Sci (2011); 1225: 59–71
- Santos et al., Brain Res (2011); 1380: 206 – 17