Example 3 Given positive integers $a, b, c$ satisfying $a<b<c$, and $ab + bc + ca = abc$. Find all $a, b, c$ that meet the conditions.

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Solve: 
$$
\begin{array}{l}
1 \leqslant a<b<c \\
\Rightarrow a b c=a b+b c+c a<3 b c \\
\Rightarrow a<3 \Rightarrow a=1 \text { or } 2 .
\end{array}
$$
(1) When $a=1$, $b+b c+c=b c$. Then $b+c=0$, which contradicts $b, c$ being positive integers.
(2) When $a=2$, $2 b+b c+2 c=2 b c$.

Then $(b-2)(c-2)=4$.
By $2<b<c$, we get
$$
\begin{array}{l}
0<b-2<c-2 \\
\Rightarrow b-2=1, c-2=4 \\
\Rightarrow b=3, c=6 .
\end{array}
$$

Therefore, the only set of positive integers that meet the conditions is
$$
a=2, b=3, c=6 \text {. }
$$