How can I execute a set of .SQL files from within SSMS?

Use SqlCmd.exe.

For example:

sqlcmd -S myServer\instanceName -i C:\myScript.sql

or to save output to a file:

sqlcmd -S myServer\instanceName -i C:\myScript.sql -o C:\EmpAdds.txt

You can execute T-SQL files without using SQLCMD Mode or using SQLCMD outside SSMS, like this:

exec sp_configure 'show advanced options',1;reconfigure with override;
exec sp_configure 'xp_cmdshell',1;reconfigure with override;

declare @sqlfile nvarchar(100),@sqlcmd varchar(4000)

exec xp_cmdshell 'del c:\SQLscript.sql && echo select ''This Is a script running on SQLCMD from SSMS'' >> c:\SQLscript.sql',no_output --test script

set @sqlfile = 'c:\SQLscript.sql' --script location
set @sqlcmd = 'sqlcmd -E -i '+@sqlfile 

exec xp_cmdshell @sqlcmd --executing script

exec sp_configure 'xp_cmdshell',0;reconfigure with override;
exec sp_configure 'show advanced options',0;reconfigure with override;

This is an addendum to Poseidon's answer:

You may have cause to preserve the state of 'xp_cmdshell' and even 'show advanced options' configuration settings, such that if they were on or off beforehand, they are toggled after (rather than always toggling them off). Here's a pair of (overdesigned) blocks I've written that will preserve the state of both as temp tables, then if they exist it will set the settings back afterwards. This has the advantage of being immune to a GO, so it will persist through the session and multiple batches. These are local temp tables though, so it will have to be the same session checking the AFTER block. You can either wrap your script with this as bookends or just run the before block, do what you need to do (in the same window or another one, the setting is server-wide), then in the same session run the after block before closing out.

-- BEFORE
IF (SELECT TOP 1 CAST([value] AS INT) FROM sys.configurations WHERE [name] = 'xp_cmdshell') <> 1
        -- xp_cmdshell is configured off, need to reconfigure to run
    BEGIN
        IF(OBJECT_ID('tempdb.dbo.#xp_cmdshell_disabled') IS NOT NULL)
            DROP TABLE #xp_cmdshell_disabled
        CREATE TABLE #xp_cmdshell_disabled (placeholder bit)

        IF (SELECT TOP 1 CAST([value] AS int) FROM sys.configurations WHERE [name] = 'Show Advanced Options') <> 1
            -- advanced options is off, must be reconfigured in order to enable xp_cmdshell
            BEGIN
                IF(OBJECT_ID('tempdb.dbo.#advanced_options_disabled') IS NOT NULL)
                    DROP TABLE #advanced_options_disabled
                CREATE TABLE #advanced_options_disabled (placeholder bit)
            
                EXEC sp_configure 'Show Advanced Options', 1
                RECONFIGURE
            END

        EXEC sp_configure 'xp_cmdshell', 1
        RECONFIGURE
    END

-- AFTER
BEGIN
    IF(OBJECT_ID('tempdb.dbo.#xp_cmdshell_disabled') IS NOT NULL)
    -- xp_cmdshell was previously off, let's turn it back off per site configuration.
        BEGIN
            EXEC sp_configure 'xp_cmdshell', 0
            RECONFIGURE
            IF(OBJECT_ID('tempdb.dbo.#xp_cmdshell_disabled') IS NOT NULL)
                DROP TABLE #xp_cmdshell_disabled
            IF(OBJECT_ID('tempdb.dbo.#advanced_options_disabled') IS NOT NULL)
                BEGIN
                    EXEC sp_configure 'Show Advanced Options', 0
                    RECONFIGURE
                    IF(OBJECT_ID('tempdb.dbo.#advanced_options_disabled') IS NOT NULL)
                        DROP TABLE #advanced_options_disabled
                END
        END
END

I had a very large INSERT and was tryin to run with help from SQLCMD, I needed to edit the file first so opened it in VS Code and was able to run INSERT without issue or hassle.

I'll be avoiding SQLCMD another ten years if I can help it