Let $a\in U(t)$. You are looking for an $x$ such that $\begin{align*} x&\equiv 1 &&\pmod{s}\\ x&\equiv a &&\pmod{t}. \end{align*}$ The fact that such an $x$ exists and is unique modulo $st$ is a consequence of the Chinese Remainder Theorem. Since $\gcd(a,t)=1$ and $\gcd(1,s)=1$, then it also follows that $\gcd(x,st)=1$, so $x\in U(st)$, as desired.
Added. The OP wants an explicit description of $x$; this is given by the Chinese Remainder Theorem, whose proof is usually constructive.
Since $\gcd(s,t)=1$, there exist integers $\alpha,\beta$ such that $\alpha s+\beta t = 1$. Then $\alpha s \equiv 0\pmod{s}$, $\alpha s\equiv 1 \pmod{t}$, $\beta t\equiv 1\pmod{s}$, and $\beta t\equiv 0\pmod{t}$. Let $x = (\alpha s)a + (\beta t)1.$ Then $x \equiv 1\pmod{s}$, and $x\equiv a\pmod{t}$, as desired.
This is the argument you can find for the proof of the Chinese Remainder Theorem in any book on elementary number theory. It's even in Wikipedia.