Take the dump from the remote RDS Server:
mysqldump -h rds.hostname -u remote_user_name -p remote_db > dump.sql
When prompted for a password, give the password for user=remote_user_name (remote server)
Upload it to your local MySQL instance:
mysql -u local_user_name -p local_db < dump.sql
Also, if you own an ec2 server in the same region, I'd suggest taking a dump there. zip the file and then scp it to your local machine. Typically, the compressed version of the file would be much smaller and you'd be able to transfer it quicker.
You can read AWS EC2 to get more information about AWS Elastic Cloud Computing.
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