In most cases, simply checking for new mail will download your messages from the server and remove those emails that were filling your quota. The best way to check what emails are still on the server is by logging into Webmail and checking how many messages are in the folders there.
If you have emails saved to separate folders in the Webmail program, or emails you sent using Webmail that were saved to “Sent Items”, those emails will apply towards your mail quota until they are removed. You will want to delete them from the server, or move them to your inbox so you can download them the next time you check for messages.
If you have just checked for new mail, and when you check Webmail you find there are messages you have downloaded still remaining in the “Inbox”, you may need to change the settings in your email software.
Check the help files of your email program to determine how to either turn on “Remove messages from server” or turn off “Leave messages on server”, depending on how your software phrases that setting.
If you do not use email software on your computer and instead handle all your email through the Webmail program, you will need to set up an email program on your computer. If you use the Webmail program exclusively, problems with the quota on your mailbox will continue to occur until you begin using email software on your computer.