First, you need to know at least one way of constructing minimum spanning trees, e.g. Kruskal's algorithm.

We claim that tunnels need to be constructed only between planets that are immediate neighbors on one of the axis. This can be easily proven. Suppose there are two planets ($A$ and $B$) and that tunnel cost between $A$ and $B$ is $|x_A-x_B|\le min\{|y_A-y_B|,|z_A-z_B|\}$. Now suppose that $A$ and $B$ are not immediate neighbors on the $X$ axis. In other words, there is some planet $C$ such that $x_A\le x_C\le x_B$. It is easy to see that instead of constructing one tunnel from $A$ to $B$ we can construct two tunnels: $(A,C)$ and $(C,B)$ that cost less than or equal to tunnel $(A,b)$ since $dist(A,C)+dist(C,B)\le |x_A-x_C|+|x_C-x_B|=|x_A-x_B|$.

We are nearly done now. We simply construct a graph of all planets with links between immediate neighbors. Using Kruskal's algorithm, we obtain a minimal cost fully connected network.