A torsion-free divisible module over a commutative integral domain is injective
Following Baer's criterion, let $I\subset R$ be an ideal and $f:I\to M$ be a homomorphism. We want to extend $f$ to a homomorphism $\bar{f}:R\to M$. To define a homomorphism $\bar{f}(r)$, you just have to pick an element $m\in M$ to be $\bar{f}(1)$ and then define $\bar{f}(r)=rm$ for each $r\in R$. For $\bar{f}$ to be an extension of $f$, you need to have $im=f(i)$ for each $i\in I$.
So you want to find $m\in M$ such that $im=f(i)$ for all $i\in I$. The condition that $M$ is divisible suggests a way to find such an $m$: just pick some nonzero $i\in I$, and then divisiblity of $I$ guarantees there exists $m\in M$ such $im=f(i)$. (This step requires $I\neq 0$; I'll let you figure out how to handle the case $I=0$ separately.)
Now of course, you only know that $im=f(i)$ for the particular $i\in I$ you chose. You still need to check that $jm=f(j)$ for all other $j\in I$. You might think it's not reasonable to expect this to be true--maybe we picked the wrong $m$. However, since $M$ is torsion-free, there is only one possible $m$ such that $im=f(i)$ (since $im=f(i)=im'$ implies $m=m'$). So if there is any $m$ that works, it must be the one we chose!
So see if you can prove that $jm=f(j)$ for all $j\in I$, using the fact that $M$ is torsion-free. The details are hidden below.
Note that $ijm=j(im)=jf(i)=f(ij)=if(j)$. Since $i\neq 0$ and $M$ is torsion-free, we can cancel $i$ and conclude that $jm=f(j)$.
There is an extremely direct proof by contradiction. Let $E(M)$ be the injective envelope of $M$, and suppose $x\in E(M)\setminus M$.
There exists an $r\in R$ such that $xr\in M\setminus\{0\}$ (since $E(M)$ is an essential extension.)
There exists a $y\in M$ such that $yr=xr$ (since $M$ is divisible.)
$x=y$ (since $M$ is torsion-free.)
This is a contradiction since $x\notin M$. Therefore $M=E(M)$. QED