How to correctly group output from multiple tables

The easiest way is to have a calendar table, defined in the following way:

CREATE TABLE calendar
(
    a_day date PRIMARY KEY
) ;

... and filled with all the relevant dates (i.e.: all days from 1990-1-1 to 2100-12-31). For the sake of simplicity, we will fill it only with year 2013:

INSERT INTO 
     calendar (a_day)
VALUES
    ('2013-01-01'),
    ('2013-01-02'),
    ('2013-01-03'),
    ('2013-01-04'),
    ('2013-01-05'),
    -- everything up to
    ('2013-12-31') ;

At this point, you can just have a JOIN with the two tables; with the join condition not being an equality, but a range condition:

SELECT
     t.id, c.a_day
FROM
     t
     JOIN calendar c ON c.a_day BETWEEN t.start_date AND t.end_date 
ORDER BY
     t.id, c.a_day ;

... and get

id | a_day     
-: | :---------
 1 | 2013-01-14
 1 | 2013-01-15
 1 | 2013-01-16
 1 | 2013-01-17
 1 | 2013-01-18
 2 | 2013-02-01
 2 | 2013-02-02
 2 | 2013-02-03
 2 | 2013-02-04

You can see all the setup at dbfiddle here

As inspired by a_horse_with_no_name, I moved the right part of the join to a subquery so the where clause doesn't interfere with the join condition

This works as intended:

WITH dates  as (SELECT min(date) as start_date,
                       max(date) as end_date
                from training_training
                where athlete_id = 1)

SELECT  distinct(d.date),
   sum(distance) OVER (ORDER BY d.date)
FROM
  (
    SELECT generate_series(start_date, end_date, interval '1 day') as date
    FROM dates
  ) d
LEFT JOIN
  (
    SELECT
      date,
      distance
    FROM training_training
    WHERE athlete_id = 1
          AND kind IN ('t', 'd', 'i', 'w')
  ) t
ON d.date = t.date

GROUP BY d.date, distance
ORDER BY d.date