Generating a list of two-dim. points from list of higher-dim. ones

Building up the answer step by step is sometimes useful. Especially if you might alter the procedure later. The pipeline below builds step by step from left to right. (Range@Length@s is same as Range[Length[s]]). You have to understand // and short pure functions to understand the line below.

Riffle[Range@Length@s, s] // Partition[#, 2] & // Map[Thread, #] & // 
 Flatten[#, 1] &

You can start just the part before the first // and then add steps sequentially to understand what is going on. That's how I developed it---from left to right.

So first

Riffle[Range@Length@s, s]

Then

 Riffle[Range@Length@s, s] // Partition[#, 2] &

And so on. Hope this is helpful.

Maybe this way:

Join @@ MapIndexed[Thread[{#2[[1]], #1}] &, s]

{{1, 1}, {1, 2}, {1, 4}, {2, 7}, {2, 11}, {3, 3}, {3, 4}, {3, 12}}

Just in case you actually want to assemble a SparseArray with "AdjacencyLists" given by s, you can do it like this:

A = With[{ci = Partition[Join @@ s, 1]},
   SparseArray @@ {Automatic, {Length[s], Max[s]}, 0., {1, {
       Prepend[Accumulate[Length /@ s], 0],
       ci
       }, ConstantArray[1, Length[ci]]}}
   ];

A["AdjacencyLists"] == s
A["NonzeroPositions"]

True

{{1, 1}, {1, 2}, {1, 4}, {2, 7}, {2, 11}, {3, 3}, {3, 4}, {3, 12}}

(And no, this use of SparseArray is not documented.)