Margin trading means that you pay a small fee to essentially borrow someone else's money to invest.
http://www.investopedia.com/university/margin/So you might be able to buy $10 worth of stocks after only putting in $1 of your own money, borrowing the other $9. This is great if the stock prices go up, if the value of everything is $15 when you sell then you can pay the lender their $9 back and keep $6 for yourself. Instead of a 50% return ($10 to $15) you have managed to make it a 600% return ($1 to $6). However if stock prices go down it's terrible. If stocks are at $5 when you sell, say, then not only did you completely loose your $1, you still owe the guy you borrowed from the other $4 that you have to give him from somewhere else. So in this example you've turned a 50% loss ($10 to $5) into a 500% loss ($1 to -$4). Margin investing is very high-risk.
What that guy is proposing is that you take $10k of your money and stick it in the stock market. You then would essentially be using it as collateral to borrow $5k to pay off your debt. Normally if you used $5k to pay of your debt and invested the other $5k then you get interest / losses on the $5k that you've invested. In this case you still have paid off your debt and have $10k gaining interest, but also have a $5k debt from borrowing on the margin.
The only reason you'd want to do it is because the interest rate from the $5k you borrowed on the margin is lower than the interest rate of your original debt. In his post he uses 6.8% for the original interest rate, and 1.6% as the margin interest rate. So in this particular case the investor would be saving 5.2% interest on his $5k debt, and adding to his life some additional risk of making a few more bucks if the stock market goes up or loosing a few more bucks if it goes down.
