It's because applying N to the whole Manipulate also causes the boundaries and the step for n to turn from integers into reals. You can see the same complex results if you evaluate the following:
Table[(7 Sin[105 n/\[Pi]]*(-1)^n)/n + 10, {n, 1, 13, 1.0}]
The only thing "special" about Manipulate (compared to Table) is that it doesn't resolve the step specification immediately, and that's why it's affected by the outer N.
If you apply N only inside the Manipulate body, you get only real values:
Manipulate[(7 Sin[105 n/\[Pi]]*(-1)^n)/n + 10 // N, {n, 1, 13, 1}]
BTW, note that the imaginary parts you see are not much more than rounding errors. You could also get rid of them using Chop.
