Chained [0][0][0][1] ends inside a String

TL;DR

Each [...] applies to the result of the previous one, and the chain changes type midway. words[0] is an Array, but words[0][0] is the String "abc". From there String#[] takes over: "abc"[0] is "a", and "a"[1] asks for the second character of a one-character string, which is nil.

Step by step

words = [["abc", ["def"]], "ghi"]

words[0]          # => ["abc", ["def"]]
words[0][0]       # => "abc"
words[0][0][0]    # => "a"
words[0][0][0][1] # => nil

The trap is the silent switch of receiver class. The first two [] are Array#[]; the last two are String#[], which indexes characters and returns one-character strings:

"abc"[0] # => "a"
"abc"[1] # => "b"
"abc"[3] # => nil

The distractor answers correspond to different chains:

words = [["abc", ["def"]], "ghi"]

words[1]    # => "ghi"
words[0][1] # => ["def"]

And "b" would require indexing "abc"[1], not "abc"[0][1].

Edge cases

Out-of-range indexing returns nil for both arrays and strings, with one classic exception: a range that starts exactly at the end returns an empty slice:

"abc"[3]   # => nil
"abc"[3..] # => ""
"abc"[4..] # => nil

[1, 2, 3][3]   # => nil
[1, 2, 3][3..] # => []

Array#dig walks nested collections, but it does not silently cross into strings; it raises instead, which turns a wrong assumption about the nesting into a loud error:

words = [["abc", ["def"]], "ghi"]

words.dig(0, 0)    # => "abc"
words.dig(0, 1, 0) # => "def"

begin
  words.dig(0, 0, 0, 1)
rescue TypeError => e
  e.message # => "String does not have #dig method"
end

Version notes

In Ruby 1.8, "abc"[0] returned the character code 97. Since Ruby 1.9, String#[] with an Integer index returns a one-character string, which is what makes this chain produce "a" and then nil.