Prove that if $p$ is a prime, then $p$ is a factor of $\binom{p}{r}$ for $r=1,2,\dots,p-1$ by using induction.
Rewrite as $(k+1)\binom{p}{k+1}=(p-k)\binom{p}{k}$.
Note that $p$ divides the right-hand side, and does not divide $k+1$, so it must divide $\binom{p}{k+1}$.
You are off to a good start, here is a hint for how to continue.
Multiply your equation by $k+1$ to give $$(k+1)\binom{p}{k+1}=(p-k)\binom pk\ .$$ By assumption $p\mid RHS$, so $p\mid LHS$. But $p\not\mid k+1$ because . . . and therefore $$p\mid\binom{p}{k+1}$$ because . . .
See if you can fill in the reasons indicated by dots.