Combining lists with respect according to length of sublists

If your first list has the form Range[n] then

b = {{1}, {1}, {1, 2}, {1, 3}, {1, 3, 4}, {1, 3, 5}};
Flatten[MapIndexed[{First[#2], #} &, b, {2}], {{2}, {1}}]

{{{1, 1}, {2, 1}, {3, 1}, {4, 1}, {5, 1}, {6, 1}},  
 {{3, 2}, {4, 3}, {5, 3}, {6, 3}},  
 {{5, 4}, {6, 5}}}

Alternatively:

a = {1, 2, 3, 4, 5, 6};
Flatten[MapThread[Thread@*List, {a, b}], {{2}, {1}}]

a = {1, 2, 3, 4, 5, 6};
b = {{1}, {1}, {1, 2}, {1, 3}, {1, 3, 4}, {1, 3, 5}};

Inner[Thread[{##}] &, a, b, Flatten[{##}, {{2}, {1}}] &]

{{{1, 1}, {2, 1}, {3, 1}, {4, 1}, {5, 1}, {6, 1}},
{{3, 2}, {4, 3}, {5, 3}, {6, 3}},
{{5, 4}, {6, 5}}}

Also

f1 = Flatten[#, {{2}, {1}}] &;
f2 = Thread /@ # &;
Composition[f1, f2, f1]@{a, b}

{{{1, 1}, {2, 1}, {3, 1}, {4, 1}, {5, 1}, {6, 1}},
{{3, 2}, {4, 3}, {5, 3}, {6, 3}},
{{5, 4}, {6, 5}}}