MIN / MAX values with Date
Your query is invalid SQL, but MySQL accepts it with the default settings. It will give you a random result. You can change this behavior by adding 'ONLY_FULL_GROUP_BY' to sql_mode.

EDIT: Similar to @MickyT's solution but using a join instead:

SELECT M.*
FROM WEATHER_MEASUREMENT M
JOIN (
   SELECT MONTH(CREATED) AS MONTH
        , MIN(AMBIENT_TEMPERATURE) AS MIN_TEMP
        , MAX(AMBIENT_TEMPERATURE) AS MAX_TEMP
   FROM WEATHER_MEASUREMENT M
   GROUP BY MONTH(CREATED)
) X
    ON MONTH(M.CREATED) = X.MONTH
   AND M.AMBIENT_TEMPERATURE IN (X.MIN_TEMP, X.MAX_TEMP);


It is not clear what time that should be used in case of a draw, the query outputs all times where a min or max temp occurred. Assuming that you would like the first occurrence, something like:

SELECT MIN(M.CREATED), M.AMBIENT_TEMPERATURE
FROM WEATHER_MEASUREMENT M
JOIN (
   SELECT MONTH(CREATED) AS MONTH
        , MIN(AMBIENT_TEMPERATURE) AS MIN_TEMP
        , MAX(AMBIENT_TEMPERATURE) AS MAX_TEMP
   FROM WEATHER_MEASUREMENT M
   GROUP BY MONTH(CREATED)
) X
    ON MONTH(M.CREATED) = X.MONTH
   AND M.AMBIENT_TEMPERATURE IN (X.MIN_TEMP, X.MAX_TEMP);
GROUP BY M.AMBIENT_TEMPERATURE


can be used.

I left out the predicate for the current month, but that is easily added. Since MySQL does not support LATERAL join, it is probably better to add that predicate to the derived table as well. I.e.

SELECT MIN(M.CREATED), M.AMBIENT_TEMPERATURE
FROM WEATHER_MEASUREMENT M
JOIN (
   SELECT MONTH(CREATED) AS MONTH
        , MIN(AMBIENT_TEMPERATURE) AS MIN_TEMP
        , MAX(AMBIENT_TEMPERATURE) AS MAX_TEMP
   FROM WEATHER_MEASUREMENT M
   WHERE (MONTH(DATE(CREATED)) = Month(NOW()))
   GROUP BY MONTH(CREATED)
) X
    ON MONTH(M.CREATED) = X.MONTH
   AND M.AMBIENT_TEMPERATURE IN (X.MIN_TEMP, X.MAX_TEMP);
WHERE (MONTH(DATE(CREATED)) = Month(NOW()))
GROUP BY M.AMBIENT_TEMPERATURE


As @RolandoMySQLDBA points out the WHERE clause can be rewritten in a index friendlier way.