`g` is a 'helper' `Dfn` (**D**efined **f**unctio**n**). It takes the length of the (right) argument and mixes the vector `aeiou` to itself that many times, yielding every possible combination of `[2, length(right arg)]` vowels.

`{⊃,/⌽(,\⌽)¨,\⌽⍵}` is another 'helper' `Dfn`. It takes the right argument and yields every substring in order.

(Thanks to @Adám for providing the correct form of the above functions)

Finally, we have:

    {((⊆g⍵)∊{⊃,/⌽(,\⌽)¨,\⌽⍵}⍵)⊆g⍵} ⍝ Main function; argument ⍵ = 'abeoic'
    {((⊆g⍵)                   )   } ⍝ Monadic ⊆ encloses the results of g⍵ so they're interpreted as vectors instead of scalars.
           ∊                        ⍝ Dyadic ∊ returns 1 for every index of the left argument that is in the right argument, and 0 otherwise.
            {⊃,/⌽(,\⌽)¨,\⌽⍵}⍵       ⍝ yields every substring of 'abeoic', in order.
                              ⊆g⍵   ⍝ Dyadic ⊆ partitions the results of it's right argument g⍵ according to a boolean (in APL, 1 for true and 0 for false) vector, which is what the dyadic ∊ returns.