How I can use like operator following by sub query?

The way I understand it is that the dependency should be a relation of the employee table to itself (a many-to-many relation).

TABLE employee
   SSN,   -- the id of the employee
   Fname -- the name of the employee

TABLE dependant
   ResponsibleSSN, -- the id of the employee being depended on
   DependantSSN    -- the id of the employee being dependant

In such a context the dependant table should not hold any name, just a foreign key to the proper employee in the employee table.

A request for your requirement could then be

SELECT DISTINCT
  e.FName
FROM
 employee e
 INNER JOIN dependant d on d.DependantSSN = e.SSN

Using LIKE in this context would mean that you know in advance the value that you are searching for, and does not make really sense.
Or maybe, I misunderstood your goal entirely?

This works, but will only return one name sadly:

SELECT [Dependent_name]
FROM [dbo].[Dependent]
WHERE [Dependent_name] LIKE 
  '%' + (SELECT TOP 1 [Fname] FROM [dbo].[Employee]) + '%'

Another option would be to use Exists:

SELECT [Fname]
FROM [dbo].[Employee]
WHERE Exists
(
    SELECT 1
    FROM [dbo].[Dependent]
    WHERE [Dependent].[Dependent_name] LIKE '%' + [Employee].[Fname] + '%'
)

EXISTS (Transact-SQL) - SQL Server | Microsoft Docs[^]