Использование:
mydata = [{'ID' : '10', 'Entry Date': '10/10/2016', 'Exit Date': '15/10/2016'},
{'ID' : '20', 'Entry Date': '10/10/2016', 'Exit Date': '18/10/2016'}]
df = pd.DataFrame(mydata)
#convert columns to datetimes
df[['Entry Date','Exit Date']] = df[['Entry Date','Exit Date']].apply(pd.to_datetime)
#repeat index by difference of dates
df = df.loc[df.index.repeat((df['Exit Date'] - df['Entry Date']).dt.days + 1)]
#add counter duplicated rows to day timedeltas to new column
df['Date'] = df['Entry Date'] + pd.to_timedelta(df.groupby(level=0).cumcount(), unit='d')
#default RangeIndex
df = df.reset_index(drop=True)
print (df)
Entry Date Exit Date ID Date
0 2016-10-10 2016-10-15 10 2016-10-10
1 2016-10-10 2016-10-15 10 2016-10-11
2 2016-10-10 2016-10-15 10 2016-10-12
3 2016-10-10 2016-10-15 10 2016-10-13
4 2016-10-10 2016-10-15 10 2016-10-14
5 2016-10-10 2016-10-15 10 2016-10-15
6 2016-10-10 2016-10-18 20 2016-10-10
7 2016-10-10 2016-10-18 20 2016-10-11
8 2016-10-10 2016-10-18 20 2016-10-12
9 2016-10-10 2016-10-18 20 2016-10-13
10 2016-10-10 2016-10-18 20 2016-10-14
11 2016-10-10 2016-10-18 20 2016-10-15
12 2016-10-10 2016-10-18 20 2016-10-16
13 2016-10-10 2016-10-18 20 2016-10-17
14 2016-10-10 2016-10-18 20 2016-10-18