At least in the contexts that I am familiar with, a family $X \to S$ with $S$ a smooth
curve is called semistable at a closed point $s$ if it has a local model (local in the analytic topology if we are working over $\mathbb C$, or local in the etale topology in general) of the form Spec $k[X_1,\ldots,X_n,t]/(X_1\ldots X_m -t)$ for some $m \leq n,$ where $t$ is a uniformizing paramater of the curves $S$ at $s$.

Note that such a local model is regular, and so if $X$ is semistable at every
point of $S$ then it is smooth.

I believe that semistability in this sense can also be characterized as follows: (i) $X$ is regular; (ii) the fibres of $X$ over points of $S$ are reduced normal crossings divisors.

This is presumably why it is required that $X$ is smooth in your context; if $X$ were not smooth, and $Y$ was a "semistable reduction" for $X$ which didn't resolve the singularities of $X$, then $Y$ could not actually be semistable in the usual sense.

A good example to think about, in the context of a family of curves, is a singularity of the following kind: Spec $k[x,y,t]/(x y = t^{n+1})$. This has a reduced normal crossings divisor over $t = 0$, but is not regular if $n > 0$. It is easy to resolve the singularity, though: just repeatedly blow-up the singular point. After you do this $n$ times, you will resolve the singularity, and you will get a reduced normal crossings divisor with $n + 2$ components: the original two components, plus $n$ components (each a $\mathbb P^1$) obtained from the blow-ups.

[Added to answer a question in the comments: If $X$ is singular, say with isolated singularities, with all the singular points lying in the singular fibres, then when we resolve the singularities of $X$, $X$ does not change in the complement of its singular points, and the singular points get replaced by various exceptional divisors. So no singular fibres will be added, but the component structure of the singular fibres will be changed.]