Assuming you already have the appropriate Akamai credentials, you should have some information like this:
Primary: rtmp://p.ep12345.i.akamaientrypoint.net/EntryPoint
Backup: rtmp://b.ep12345.i.akamaientrypoint.net/EntryPoint
User: ######
Pass: ******
Stream: [EVENT_ANGLE_BITRATE]@12345
Playback: http://zencoder_hd1_1-lh.akamaihd.net/[EVENT_ANGLE_BITRATE]@12345
Open up FMLE and put the primary, backup and stream information in. Your stream name should look something like awesomestream_01_600@12345. Akamai's best-practices suggest the EVENT_ANGLE_BITRATE format, but you will not be able to connect if you forget to add the @12345.
Once you click connect you should get a username and password box.
Put in the username and password provided by Akamai and press OK. The same dialog should pop up again, this time for the backup url. After you've done this, double check your output settings to make sure they're correct. Keep in mind when doing this that what you output is exactly what users will be accessing, so you might want to be fairly conservative with bandwidth consumption. Once you're ready click "Start" and you should then be streaming.
You can test your stream by entering the playback url plus the stream name into Akamai's test player. Using the example data from the beginning, it should look something like http://zencoder_hd1_1-lh.akamaihd.net/awesomestream_01_600@12345.