The problem is not the JSONArray.toString(), as @Selvin mentioned.

From JSONArray source:

/**
 * Encodes this array as a compact JSON string, such as:
 * <pre>[94043,90210]</pre>
 */
@Override public String toString() {
    try {
        JSONStringer stringer = new JSONStringer();
        writeTo(stringer);
        return stringer.toString();
    } catch (JSONException e) {
        return null;
    }
}

/**
 * Encodes this array as a human readable JSON string for debugging, such
 * as:
 * <pre>
 * [
 *     94043,
 *     90210
 * ]</pre>
 *
 * @param indentSpaces the number of spaces to indent for each level of
 *     nesting.
 */
public String toString(int indentSpaces) throws JSONException {
    JSONStringer stringer = new JSONStringer(indentSpaces);
    writeTo(stringer);
    return stringer.toString();
}

The problem is that you need to convert your ChecklistAnswer to JSON object first for your JSONArray to work properly.

Again from Javadoc:

/**
 * A dense indexed sequence of values. Values may be any mix of
 * {@link JSONObject JSONObjects}, other {@link JSONArray JSONArrays}, Strings,
 * Booleans, Integers, Longs, Doubles, {@code null} or {@link JSONObject#NULL}.
 * Values may not be {@link Double#isNaN() NaNs}, {@link Double#isInfinite()
 * infinities}, or of any type not listed here.

...

Take a look at this tutorial. Also you can parse above json like :

JSONArray arr = new JSONArray(yourJSONresponse);
List<String> list = new ArrayList<String>();
for(int i = 0; i < arr.length(); i++){
    list.add(arr.getJSONObject(i).getString("name"));
}

As long as it's a valid JSONArray to begin with, then yes.

Here you get JSONObject so change this line:

JSONArray jsonArray = new JSONArray(readlocationFeed); 

with following:

JSONObject jsnobject = new JSONObject(readlocationFeed);

and after

JSONArray jsonArray = jsnobject.getJSONArray("locations");
for (int i = 0; i < jsonArray.length(); i++) {
    JSONObject explrObject = jsonArray.getJSONObject(i);
}

You have a JSONArray like this:

[
    "my string"
]

Usually, JSONArrays are used - like normal arrays - with multiple entries. You are currently converting the whole JSONArray to a string. The \ is there because without it, the quotes would terminate that string.

What you want is to get only the first element out of that array. If you know that your array contains only one string, you can do something like

String result = myJSONArray.getString(0);