É um operador de correspondência de expressão regular.
Na página bash
man:
An additional binary operator, =~, is available, with the same
precedence as == and !=. When it is used, the string to the
right of the operator is considered an extended regular
expression and matched accordingly (as in regex(3)). The return
value is 0 if the string matches the pattern, and 1 otherwise.
Veja a man page de bash
para mais detalhes (procure por =~
)