What you have is a digital indicator with a step-width of 0.1 s.
When you look in the JCGM 100 from the BIPM "Evaluation of measurement data — Guide to the expression of uncertainty in measurement", this is called a Type B evaluation of standard uncertainty (see chapter 4.3). If we assume that the value is rounded to the nearest 0.1 s, then we can assume a rectangular probabilty distribution with a half-width of 0.05 s (see chapter 4.3.7 and 4.4.5). This means that the probability of the real value to be in between [15.65, 15.74] is 100 % and 0 % percent outside of that.
The best estimate for the standard uncertainty given in chapter for 4.3.7 for such a case is $$u = a/\sqrt{3}$$ with 95% coverage and $a$ being the half-width of the step width of the stop watch. This gives $$u=0.05s/\sqrt{3}=0,029s$$ or $0,37\,\%$ relative.
As for the significant figures with the uncertainty, you would always include the two first digits other than zero (sorry for this english). Lets say you have $u=0,0289s$, then you print it as $u=0,029s$ and you include that amount of figures in the result as well, so you would write for your example $t=15.700s \pm 0.029s$ or just $t=15.700(29)s$.
Since this result differs from the correct answer, I would really be interested in the reasoning behind it, if you have the chance to ask the person responsible.
Measurement uncertainty calculation is a very complex topic btw! In the quoted document are lots of examples for different situations that might help you with future problems!
Links:
https://www.bipm.org/en/committees/jc/jcgm/publications
https://www.bipm.org/documents/20126/2071204/JCGM_100_2008_E.pdf/cb0ef43f-baa5-11cf-3f85-4dcd86f77bd6)