I am a bit confused when it says "this has semiannual compounding because payments are assumed to be made every 6 months. With continuous compounding, the rate is 6.75% per annum." Isn't the rate of 6.87% already assuming continuous compounding and is obtained by solving the equation here? What does it mean with continuous compounding, the rate is 6.75% per annum? Where is this coming from? And when it says "this has semiannual compounding", what is it referring to?
The way I like to explain this is with a notion of quoting. It's a convention to quote the coupons annualized by multiplying them by frequency. Suppose, the coupon is semiannual and equal to 3.375% of the outstanding. This is how much interest is accrued during 6 months. However, it is the convention to quote it on annualized basis, i.e. multiplied by 2 since it's semiannual. So, the coupon is quoted as c=6.75%.
Compounded, this coupon will yield the following in one year: $$(1+c/2)^2=1+y$$ $$y=(1+c/2)^2-1\approx 6.86\%$$
Alternatively you can quote the same yield on continuous compounding base: $$(1+c/2)^2=e^y$$ $$y=2\ln (1+c/2)\approx 6.64\%$$