I am actually doing this now. It's a lot of work to get it started, but once started the money comes once a month. First, you must attend a VA approved Part 141 flight school. After you find one of those the individual who handles the VA students/paperwork for the school will instruct you to go onto the internet to the VA's gi bill website and look for a fun document called the VEAP. This online document takes a little while, but once you complete it, you must submit it electronically, print out the signature page and mail it into the nearest VA.
Now you wait. The wait can be a week or a couple of months. You will eventually receive a letter from the VA saying that your VEAP has been processed, and approved. You must take the approval letter to the school.
Now while all this is taking place you CAN began flight training. Once you receive your approval the school can go back up to a full calendar year to get you reimbursed. Reimbursement includes everything! Ground school, pre and posts, actual flight time (both dual and solo.)
Once a month your school should present you with a form that lays out all the charges you incurred for the month. At the bottom of the form there will be a payment amount. This should equal 60% of your costs for the month.
That's basically it. Just a couple of side notes though, the multi-engine syllabus must be Part 141 VA approved as well as the school. You must also have a current 1st or 2nd class medical. No 3rd class medicals. And for whatever reason they want you to get a new medical every year. I know, I know they are good for longer, but it's the VA's requirement.
This guidance works for ANY advanced ratings as well. As long you are not trying to get you Recreational Lic, Sport Lic, or PPL you can get reimbursed as I outlined above. Hope this helps.