Here's one way to do it. The concept is to first merge the groups, then find the missing groups.

To merge the groups, the basic idea is:

1. Split the ranges so you have every possible sub-range. e.g. 25 Jul - 19 Sep becomes 25 Jul - 11 Aug, 11 Aug - 17 Aug, 17 Aug - 21 Aug, etc.
2. Group together the consecutive ranges using the Tabibitosan method

You can do step 1 by:

- Unpivoting the original dataset so you have a single column of dates. Each of these is the start of a range
- The end of each range is the next row. So you can calculate this with lead
- Note you'll need to squash out duplicate start dates at this point! You can do this by calculating the dense_rank of start dates on the original set. Then group by this value on the unpivoted rows.

You now have a set of ranges.

alter session set nls_date_format = 'dd-Mon-yy';
create table t (
  stdt date,
  endt date
);
insert into t values ('15-Apr-16', '29-Apr-16');
insert into t values ('29-Apr-16', '30-Apr-16');
insert into t values ('09-May-16', '12-May-16');
insert into t values ('10-May-16', '12-May-16');
insert into t values ('25-Jul-16', '19-Sep-16');
insert into t values ('11-Aug-16', '17-Aug-16');
insert into t values ('21-Aug-16', '05-Sep-16');
insert into t values ('28-Aug-16', '17-Sep-16');
insert into t values ('06-Sep-16', '10-Oct-16');
commit;

with ranges as (
  select c1, dt, lead(dt) over (order by dt) ld, 
         dense_rank() over (order by dt) dr
  from   t
  unpivot (dt for c1 in (stdt, endt))
), grps as (
  select min(dt) st, max(ld) en, dr
  from   t, ranges r
  where  t.stdt <= r.dt and r.dt < t.endt
  group  by dr
)
  select * from grps
  order  by dr;

ST         EN         DR  
15-Apr-16  29-Apr-16  1   
29-Apr-16  30-Apr-16  2   
09-May-16  10-May-16  4   
10-May-16  12-May-16  5   
25-Jul-16  11-Aug-16  7   
11-Aug-16  17-Aug-16  8   
17-Aug-16  21-Aug-16  9   
21-Aug-16  28-Aug-16  10  
28-Aug-16  05-Sep-16  11  
05-Sep-16  06-Sep-16  12  
06-Sep-16  17-Sep-16  13  
17-Sep-16  19-Sep-16  14  
19-Sep-16  10-Oct-16  15

To do step 2

- Increment a counter for each row, ordered by start date. (row_number)
- Then find the difference between this at the dense_rank value
- Group by this calculation and get the min start and max end to get the consecutive groups:

with ranges as (
  select c1, dt, lead(dt) over (order by dt) ld, 
         dense_rank() over (order by dt) dr
  from   t
  unpivot (dt for c1 in (stdt, endt))
), grps as (
  select min(dt) st, max(ld) en, dr
  from   t, ranges r
  where  t.stdt <= r.dt and r.dt < t.endt
  group  by dr
), tab as (
  select st, en, 
         dr - row_number() over (order by st) grp
  from   grps
), oper as (
  select 'Operational' status, min(st) st, max(en) en from tab
  group  by grp
)
  select * from oper
  order  by 1;

STATUS       ST         EN         
Operational  09-May-16  12-May-16  
Operational  25-Jul-16  10-Oct-16  
Operational  15-Apr-16  30-Apr-16 

You now have the operational ranges.

The non-operational ranges:

- Start at the end of each operational range
- End at the start of the next operational range

So you can find this with lead again:

  select 'Non-operational' status, 
  en + 1 st, lead(st) over (order by st) - 1 en
  from   oper

To get everything in a single query, union (all) the results together and exclude the final non-operational group (that has a null end):

with ranges as (
  select c1, dt, lead(dt) over (order by dt) ld, 
         dense_rank() over (order by dt) dr
  from   t
  unpivot (dt for c1 in (stdt, endt))
), grps as (
  select min(dt) st, max(ld) en, dr
  from   t, ranges r
  where  t.stdt <= r.dt and r.dt < t.endt
  group  by dr
), tab as (
  select st, en, 
         dr - row_number() over (order by st) grp
  from   grps
), oper as (
  select 'Operational' status, min(st) st, max(en) en from tab
  group  by grp
), non_oper as (
  select 'Non-operational' status, 
        en + 1 st, lead(st) over (order by st) - 1 en
  from   oper
)
  select * from oper
  union all
  select * from non_oper
  where  en is not null
  order  by 2, 3;

STATUS           ST         EN         
Operational      15-Apr-16  30-Apr-16  
Non-operational  01-May-16  08-May-16  
Operational      09-May-16  12-May-16  
Non-operational  13-May-16  24-Jul-16  
Operational      25-Jul-16  10-Oct-16

If you want to know more about dealing with gaps and overlaps in dates, I recommend reading Stew Ashton's blog posts:

https://stewashton.wordpress.com/2014/03/11/sql-for-date-ranges-gaps-and-overlaps/