The $7,650 is the difference between your $30,000 taxable income and the upper limit of the 15% bracket, which is $37,650. $37,650 - $30,000 = $7,650. It's the amount of space you have left in the 15% bracket for additional income that would still be in that bracket.
If you add a capital gain of $7,650 to the $30,000 of ordinary income, then your total income is $37,650 and, yes, all your income is still in the 15% bracket, but just barely. That's the maximum income you can have and still stay in the 15% bracket. So you are correct that the tax rate on the capital gain would be 0%. But any additional income above the total of $37,650, including any additional capital gain, would go into the next bracket, which is the 25% bracket.
For example, suppose your capital gain was $8,650 instead of $7,650. Your total taxable income is now $38,650, which is $1,000 above the top of the 15% bracket. That additional $1,000 of capital gain would be in the 25% bracket, and would be taxed at the capital gains rate for the 25% bracket, which is 15%. The capital gain is added to your ordinary income, and the total determines what bracket you are in. The amount that falls in the highest bracket - the $1,000 in this example - is considered to be part of the capital gains. That's what's meant by "stacking" the capital gains on top of your ordinary income. The capital gain, or at least part of it, is always treated as being in the highest bracket that you hit.
