I am not sure how the math is exactly done for the interest to come out like that. Also, what is the capitalized interest?
EDIT: What is an outstanding principal balance?
The principal of the loan is the amount you borrow. The capitalized interest is added to the principal of the loan, because you are not paying this interest as it accrues. So when you begin payments, the principal of the loan is $5,500 + $436 = $5,936.
Using the standard amortization formula (see this page for details), the per-month payment for a ten-year payment plan at 6.8% interest on principal of $5,936 is $68.31. One hundred twenty payments (each month for ten years) totals $8,197.40.
"Outstanding principal balance" is the amount you owe at any given time, not including the amount of interest you need to pay as soon as possible.
The "capitalized interest" shown is consistent with an average of 13.5 months between when each dollar is borrowed and when the repayment period begins.
Suppose you borrow the first half of the money on September 1, 2017 and the second half of the money on February 1, 2017 (5 months later). At that point, half the money has been accruing interest for 5 months. On January 1, 2018, half the money accrued interest for 16 months, and half the money accrued interest for 11 months. The lender now expects you to start repaying the loan, with the first payment due at the end of January 2018 or the beginning of February 2018. If you make the minimum payments on time, the lender expects you to make 120 monthly payments. The last monthly payment would be at the end of December 2027 or the beginning of January 2028.
The lender (or the website) should provide details about the actual payment plan, grace periods, provisions for handling inability to pay due to unemployment, and other terms.
In the United States, most installment loans pretend that (for purposes of calculating interest) every month has 30 days -- even February and July! Each month, 1/12 of the "annual percentage rate" (APR) is charged as interest. If you do the compounding, a 6.8 percent APR corresponds to (1 + 0.068 / 12)^12 - 1 = 7.016 percent "annual percentage yield" (APY).
Also, the APR is understated. The 6.8 percent applies to the full balance (including the loan fees), even though the borrower only gets the amount minus the loan fees. The 6.8 percent rate is useful for doing calculations after the loan fees have been charged, though. These calculations include the capitalized interest and the monthly payment amounts.
A true calculation of the APR would take the loan fees into account, and give a higher number than 6.8 percent. But the corrected APR would not be useful for calculating the capitalized interest, nor for calculating the monthly payment amounts.