T-SQL doesn't have native regex support. You can use a CLR function to access .NET regex functionality or use PatIndex if the pattern is simple.
Or if you just want to get the contents of [...] maybe CharIndex would work.
;with strings as
(
SELECT 'no match' AS string UNION ALL
SELECT '[4888378977CA4A5] Test String' UNION ALL
SELECT 'RE: [Other Value] Test String' 
)
select substring(string,
             charindex('[',string)+1,
             charindex(']',string, charindex('[',string))-charindex('[',string)-1)
                                                                          AS result
from strings
where string like '%/[%/]%' ESCAPE '/'
Returns
result
-----------------------------
4888378977CA4A5
Other Value