Here is an alternative to Count Dracula's (correct) argument that emphasizes instead the constancy of the Hilbert polynomial for a flat family of projective schemes.
As above, assume that $Y$ is a DVR. For one fixed irreducible component $Z_{\eta}$ of $X_\eta$ of minimal dimension $d$, denote by $Z$ the Zariski closure of $Z_{\eta}$ in $X$ together with its closed immersion $u:Z\to X$. Denote by $W_{\eta}$ the union of all other irreducible components of $X_{\eta}$, and denote by $v:W\to X$ the closure of $W_\eta$ in $X$. By construction, both $Z$ and $W$ are flat over $Y$, since every associated point is a generic point of $Z_\eta$, resp. $W_\eta$.
The closed immersion $u$ and $v$ determine an associated morphism of $\mathcal{O}_X$-modules, $$(u^\#, v^\#):\mathcal{O}_X \to u_*\mathcal{O}_Z \oplus v_*\mathcal{O}_W.$$ The restriction of $(u^\#,v^\#)$ on $X_\eta$ is injective. Thus the kernel of $(u^\#,v^\#)$ is a subsheaf of $\mathcal{O}_X$ that is torsion for $\mathcal{O}_Y$. Since $\mathcal{O}_X$ is flat over $\mathcal{O}_Y$, the kernel of $(u^\#,v^\#)$ is the zero sheaf.
Denote the quotient of $(u^\#,v^\#)$ by $\mathcal{Q}$. The restriction of $\mathcal{Q}$ on $X_\eta$ has support whose dimension is strictly smaller than the dimension of any irreducible component of $X_\eta$. In particular, the Hilbert polynomial of $\mathcal{O}_{X_\eta}$ agrees with the Hilbert polynomial of $\mathcal{O}_{Z_\eta}\oplus \mathcal{O}_{W_\eta}$ modulo the subspace of numerical polynomials of degree strictly less than $d = \text{dim}(Z_\eta)$.
Now consider the restriction $(u_0^\#,v_0^\#)$ of $(u^\#,v^\#)$ to the closed fiber $X_0$. By the flatness hypothesis, the Hilbert polynomials of the domain and target of this homomorphism equal the Hilbert polynomials on the generic fiber. Thus the difference of these Hilbert polynomials on $X_0$ equals the difference of these polynomials on $X_\eta$, and we know that this difference is a polynomial of degree strictly less than $d$. The cokernel of $(u_0^\#,v_0^\#)$ equals the restriction $\mathcal{Q}_0$. If the induced morphism $\mathcal{O}_{Z_0}\to \mathcal{Q}_0$ is nonzero at some generic point of $Z_0$, then the support of $\mathcal{Q}_0$ has an irreducible component of dimension $\geq d$. Thus the Hilbert polynomial of $\mathcal{Q}_0$ has degree $\geq d$. Since the difference polynomial has degree strictly less than $d$, the kernel of $(u_0^\#,v_0^\#)$ has Hilbert polynomial of degree $\geq d$ counterbalancing the Hilbert polynomial of $\mathcal{Q}_0$. In particular, the kernel of $(u_0^\#,v_0^\#)$ is not zero.
By the flatness hypothesis, every associated point of $X$ is contained in $X_\eta$. Thus, every generic point $\xi$ of $X_0$ is the specialization of a generic point that is either in $Z$ or in $W$. Thus the localization $\mathcal{O}_{X_0,\xi}$ either factors through $\mathcal{O}_{Z_0,\xi}$ or factors through $\mathcal{O}_{W,\xi}$. Therefore the kernel of $(u_0^\#,v_0^\#)$ is in the kernel of the localization at every generic point of $X_0$. Since the kernel is nonzero, $X_0$ has embedded associated points, contradicting the hypothesis that $X_0$ is geometrically reduced. Therefore, by way of contradiction, the support of $\mathcal{Q}_0$ does not contain $Z_0$. So $Z_0$ is not contained in $W_0$.
Now we continue by induction on the number of irreducible components, replacing $X$ by $W$.
Best Answer
If you have a flat, or even better a smooth morphism, then the intuition is that there is some sort of continuity along the fibres. It is then not unreasonable to make definitions by looking at fibres. (You should think about Erhesmann's theorem, which says that a smooth proper map of smooth manifolds is a fibre bundle. So if you want to have a bundle of, say, genus g surfaces over a manifold $X$, it is enough to ask for a proper smooth map $Y \to X$ each of whose fibres is a genus $g$ surface. Similarly, switching back to the language of algebraic geometry, if we want a family of genus $g$ curves, it makes sense to ask for a smooth proper map whose fibres are curves of genus $g$.)