You can follow the following steps:
1. Download and upload all the files of the myBB into the new server. I think you already did this.
2. Export the database from phpMyadmin (and interface provided by most web servers to interact with database). This will generate a sql file. Google how to export and import a database from phpMyadmin for exact instructions.
3. Create a new database into your new server. Note down database name, username, password. Also note down database server host (this is usually 'localhost'). You can see the documentation of the new host support page for further instructions.
4. Import the sql file into the new database via phpMyadmin interface.
5. Make changes to the configuration file. Open config.php, located inside the 'inc' folder of the myBB. Replace the following things with the new database details:
$config['database']['hostname'] = 'new hostname, usually localhost';
$config['database']['username'] = 'new database username';
$config['database']['password'] = 'new database password';
$config['database']['database'] = 'new database name';
If you did everthing correctly, you should be good to go. If you struck with anything, use google. I have rougly written the steps, but for exact methods you'll have to find yourself.