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PG 9.4+

The accepted answer is definitely what you need, but for the sake of simplicity here is a helper I use for this:

CREATE OR REPLACE FUNCTION jsonb_array_to_text_array(p_input jsonb)
 RETURNS text[]
 LANGUAGE sql
 IMMUTABLE
AS $function$

SELECT array_agg(ary)::text[] FROM jsonb_array_elements_text(p_input) AS ary;

$function$;

Then just do:

SELECT jsonb_array_to_text_array('["a", "b", "c"]'::jsonb);

Updated 2/23/2020 in response to comments: Comments are correct that this could be more efficient. At the time I posted there was no modularized solution offered so I offered one in earnest, if non-optimal. Since then Erwin has updated his answer with a simple and efficient function so I never updated mine. Updating it now since there is still attention coming to this answer

PG 9.4+

The accepted answer is definitely what you need, but for the sake of simplicity here is a helper I use for this:

CREATE OR REPLACE FUNCTION jsonb_array_to_text_array(p_input jsonb)
 RETURNS text[]
 LANGUAGE sql
 IMMUTABLE
AS $function$

SELECT array_agg(ary)::text[] FROM jsonb_array_elements_text(p_input) AS ary;

$function$;

Then just do:

SELECT jsonb_array_to_text_array('["a", "b", "c"]'::jsonb);

Updated 2/23/2020 in response to comments: Comments are correct that this could be more efficient. At the time I posted there was no modularized solution offered so I offered one in earnest, if non-optimal. Since then Erwin has updated his answer with a simple and efficient function so I never updated mine. Updating it now since there is still attention coming to this answer

One more update, because this just bit me: The above function will return null if there are no values. This may not be desirable depending on your situation. Here's a function which returns an empty array if the value is not null, but still returns null if the input is null.

CREATE OR REPLACE FUNCTION jsonb_array_to_text_array_strict(p_input jsonb)
 RETURNS text[]
 LANGUAGE sql
 IMMUTABLE
AS $function$

SELECT 
  CASE 
    WHEN p_input IS null 
    THEN null 
    ELSE coalesce(ary_out, ARRAY[]::text[]) 
  END
FROM (
  SELECT array_agg(ary)::text[] AS ary_out
  FROM jsonb_array_elements_text(p_input) AS ary
) AS extracted;

$function$
;

PG 9.4+

The accepted answer is definitely what you need, but for the sake of simplicity here is a helper I use for this:

CREATE OR REPLACE FUNCTION jsonb_array_to_text_array(
  p_input jsonb
 ) RETURNS TEXT[] AS $BODY$

DECLARE v_output text[];

BEGIN

  SELECT array_agg(ary)::text[]
  INTO v_output
  FROM jsonb_array_elements_text(p_input) AS ary;

  RETURN v_output;

END;

$BODY$
LANGUAGE plpgsql VOLATILE;

Then just do:

SELECT jsonb_array_to_text_array('["a", "b", "c"]'::jsonb);

PG 9.4+

The accepted answer is definitely what you need, but for the sake of simplicity here is a helper I use for this:

CREATE OR REPLACE FUNCTION jsonb_array_to_text_array(p_input jsonb) 
 RETURNS text[]
 LANGUAGE sql
 IMMUTABLE
AS $function$

SELECT array_agg(ary)::text[] FROM jsonb_array_elements_text(p_input) AS ary;

$function$;

Then just do:

SELECT jsonb_array_to_text_array('["a", "b", "c"]'::jsonb);

Updated 2/23/2020 in response to comments: Comments are correct that this could be more efficient. At the time I posted there was no modularized solution offered so I offered one in earnest, if non-optimal. Since then Erwin has updated his answer with a simple and efficient function so I never updated mine. Updating it now since there is still attention coming to this answer

The accepted answer is definitely what you need, but for the sake of simplicity here is a helper I use for this:

CREATE OR REPLACE FUNCTION jsonb_array_to_text_array(
  p_input jsonb
) RETURNS TEXT[] AS $BODY$

DECLARE v_output text[];

BEGIN

  SELECT array_agg(ary)::text[]
  INTO v_output
  FROM jsonb_array_elements_text(p_input) AS ary;

  RETURN v_output;

END;

$BODY$
LANGUAGE plpgsql VOLATILE;

PG 9.4+

The accepted answer is definitely what you need, but for the sake of simplicity here is a helper I use for this:

CREATE OR REPLACE FUNCTION jsonb_array_to_text_array(
  p_input jsonb
) RETURNS TEXT[] AS $BODY$

DECLARE v_output text[];

BEGIN

  SELECT array_agg(ary)::text[]
  INTO v_output
  FROM jsonb_array_elements_text(p_input) AS ary;

  RETURN v_output;

END;

$BODY$
LANGUAGE plpgsql VOLATILE;

Then just do:

SELECT jsonb_array_to_text_array('["a", "b", "c"]'::jsonb);