EDIT: updated as per OP comment.
How about this:
travel_dates[] <- lapply(travel_dates, as.Date, format="%m/%d/%y")
dts <- with(travel_dates, mapply(seq, Departure, Return, by="1 day"))
This produces a list with as many items as you had rows in your initial table. You can then summarize (this will be data.frame with the number of times a date showed up):
data.frame(count=sort(table(Reduce(append, dts)), decreasing=T))
# count
# 2013-07-06 3
# 2013-07-07 3
# 2013-07-08 3
# 2013-07-09 3
# ...
OLD CODE:
The following gets the #days of each trip, rather than a list with the dates.
transform(travel_dates, days_away=Return - Departure + 1)
Which produces:
# Departure Return days_away
# 1 2013-07-06 2013-08-03 29 days
# 2 2013-07-06 2013-08-03 29 days
# 3 2013-06-28 2013-08-07 41 days
If you want to put days_away
in a separate list, that is trivial, though it seems more useful to have it as an additional column to your data frame.