Your root account, and this statement applies to any account, may only have been added with localhost access (which is recommended).
You can check this with:
# mysql -u root -p
> SELECT host FROM mysql.user WHERE User = 'root';
If you only see results with localhost and 127.0.0.1, you cannot connect from an external source. If you see other IP addresses, but not the one you're connecting from - that's also an indication.
You will need to add the IP address of each system that you want to grant access to, and then grant privileges:
The easiest way to create a user account with the mysql_native_password authentication plugin is to make sure that old_passwords=0 is set, and then create a user account via CREATE USER that does not specify an authentication plugin, but does specify a password via the IDENTIFIED BY clause.
SET old_passwords=0;
CREATE USER 'root'@'ip_address' IDENTIFIED BY 'some_pass';
GRANT ALL ON *.* TO 'root'@'ip_address';
If you see %, well then, there's another problem altogether as that is "any remote source". If however you do want any/all systems to connect via root, use the % wildcard to grant access:
CREATE USER 'root'@'%' IDENTIFIED BY 'some_pass';
GRANT ALL ON *.* TO 'root'@'%';
Finally, reload the permissions, and you should be able to have remote access:
FLUSH PRIVILEGES;