The date/time value can be either a date (as in the example here) or a more complete date/time such as 2019-02-17T17:00:00+0000 to indicate 5pm UTC on 17th February 2019.
In case it helps anyone, here’s how to get the forms submitted during a time range, such as a single-day’s submissions. – we use this for pulling yesterday’s submissions in crontab. Doubtless there are other ways, but it an be tricky to pull the pieces together from the doc, as others have also found.
# fetch token from https://kf.kobotoolbox.org/token/
TOKEN=0123456789012345678901234567890123456789
URL='https://kc.kobotoolbox.org/api/v1/data'
# form id from list at data URL
FORM=123456
# -s silent
# -S show error
# -f fail with error status
# -g no globbing, needed for braces in url
curl -sSkg \
-X GET \
-H "Authorization: Token $TOKEN" \
$URL/$FORM'?query={"$and": [{"_submission_time": {"$gte": "2019-05-28"}},{"_submission_time": {"$lt": "2019-05-29"}}] }'