It seems that it is sort of comparison operator, but what exactly it does in e.g. the following code (taken from https://github.com/lvv/git-prompt/blob/master/git-prompt.sh#
It matches regular expressions
Refer to following example from http://tldp.org/LDP/abs/html/bashver3.html#REGEXMATCHREF
#!/bin/bash
input=$1
if [[ "$input" =~ "[0-9][0-9][0-9]-[0-9][0-9]-[0-9][0-9][0-9][0-9]" ]]
# ^ NOTE: Quoting not necessary, as of version 3.2 of Bash.
# NNN-NN-NNNN (where each N is a digit).
then
echo "Social Security number."
# Process SSN.
else
echo "Not a Social Security number!"
# Or, ask for corrected input.
fi