The volume $V_n(t)$ is the cdf of the sum of $n$ independent random variables, uniformly distributed on $[0,1]$. So the density $V_n'(t)$ is the $n$-fold convolution of the characteristic function of the unit interval, $\chi_{[0,1]}$. (In particular $V_n'\in C^{n-1}(\mathbb{R})$$V_n'\in C^{n-2,1}(\mathbb{R})$, with support in $[0,n]$, and polynomial of degree less than $ n $ on any interval between consecutive integers). It is convenient to write $\chi_{[0,1]}(t)=H(t)-H(t-1)$, where $H(t):=\chi_{\mathbb{R_+}}(t)$ is the Heaviside function, or $\chi_{[0,1]}=H-\tau^1 H$ where $\tau^s:f\mapsto f(\cdot-s)$ is the translation semigroup on functions. Recall that convoluting with $H$ is taking a primitive: $(H*f)(x)=\int_{-\infty}^xf(t)dt$, so in particular $H^{*(n+1)}(t)=t_+^n/n!$, and that $\tau^s(f*g)=\tau^sf*g= f* \tau^s g$.
Thus $$V_n=H*V'_n=H*(H-\tau^1H)^{*n}=\sum_{k=0}^n(-1)^k{n\choose k}H*H^{*(n-k)}*(\tau^1H)^{*k}$$ $$=\sum_{k=0}^n(-1)^k{n\choose k} \tau^kH^{*(n+1)},$$ that is $$V_n(t)=\sum_{k=0}^n\frac{(-1)^k}{k!(n-k)!} {(t-k)_+}{^n}\, .$$