What is the Google sandbox? What happens in the Google sandbox as far as SEO is concerned?
When I submit my site to Google Seach Console, what is the process to move from the sandbox to the Google search index?
Strictly speaking there is no such thing as the Google Sandbox. Some SEO related sites will state that all new sites are placed into a sandbox which by their definitions is effectively a manner by which Google does not rank the new sites as highly as established sites on order to prevent spamming. Google does not actually sandbox new sites. Google does take into consideration the relative age of a site (based on domain registration and the age of the indexed pages) but these are relatively low signals compared to the indexing signals coming from keywords, organic links, etc.
Really what @closetnoc says is fairly accurate in that this may have been something that was being done a while back but these days Google has found that using the age of a site is not as accurate a method of ranking a site as evaluating the content of the site.
The basic thing you need to know about SEO is that you shouldn't be thinking about it as SEO. Google considers content as king and actually states that sites should be designed and optimized for users and not for search engines. A better way to put it would be to call the exercise Human Optimization and not Search Engine Optimization.