An incidence matrix of all subsets

It isn't quite clear to me what is wanted, but the matrix in the OP can be generated readily, with a little help from an undocumented function:

With[{n = 3}, 
     SparseArray`SparseBlockMatrix[MapIndexed[Join[#2, #2] -> {#1} &,
                                              Unitize[Subsets[Range[n],
                                                              {1, ∞}]]]]] // MatrixForm

$$\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ \end{pmatrix}$$

sL = Rest[Subsets@Range@3]

 {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}

You can also use SparseArray + Band:

ClearAll[f]
f = SparseArray[Band[{1, 1}] -> List /@ Unitize@#] &;


f @ sL // MatrixForm // TeXForm

$\left( \begin{array}{cccccccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ \end{array} \right)$

Alternatively, a combination of MapThread + RotateRight + PadLeft + Accumulate:

ClearAll[g]
g = Module[{cl = Accumulate[Length /@ #]}, 
  MapThread[RotateRight, {PadLeft[Unitize @ #, {Automatic, Last @ cl}], cl}]] &

g @ sL // MatrixForm // TeXForm

same result

If you wish to use an integer as input, you can modify f as follows:

ClearAll[f2]
f2 = SparseArray[Band[{1, 1}] -> List /@ Unitize @ Rest @ Subsets @ Range @ #] &;

f2 @ 3 // MatrixForm // TeXForm

$\left( \begin{array}{cccccccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ \end{array} \right)$

You can modify g similarly.

Clear["Global`*"]

n = 3;

L = Range[n];

SL = Subsets[L, {1, n}]

(* {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} *)

Freestanding commas do not make much sense. Perhaps you mean:

(mat = Module[{len = Length@SL},
    Array[ConstantArray[KroneckerDelta@##, Length[SL[[#2]]]] &, {len, 
      len}]]) // MatrixForm