The English equivalent would be:-
-- in a few days;
-- for a little while.
The idea being that the person has no definite idea when he or she will be back and hence the vagueness or "elastic" period of between 10 to 15 days.
The Chinese divides a month into 3 periods of 10 days each called a 旬 (Xún) The first 10 is 上 旬, middle 10 is 中 旬, and last 10 is下 旬. (Much like the Western way of saying 1st, 2nd, 3rd, 4th quarter of a calendar year) Thus giving an elastic period of 10 days or so to play around with because of uncertainty, hence the 十天半月.
This practice stems from a time when traveling meant walking the whole way or horse cart or river boat unlike now where one could say quite definitely when things happen.
An equivalent with a longer period of time is "一 年 半 载" (Zài), in a year or so; in about a year's time.