How to get the strings between the given string in oracle

Use INSTR to find the positions of the delimiters and then SUBSTR to extract the sub-strings:

Oracle Setup:

CREATE TABLE test_data ( value ) AS
  SELECT 'APP_APX_PLM_PostCategory~~~pavan anand~~~2019-09-26 15:03:39@@@MF_APX_PLM_PostBuyProgram.msgflow***MF_APX_PLM_PostBuyProgram1.msgflow***MF_APX_PLM_PostBuyProgram2.msgflow^^^APP_APX_PLM_Promo~~~skola2
~~~2019-09-30 14:34:11@@@MF_APX_PLM_Promo1.msgflow***MF_APX_PLM_Promo2.msgflow^^^APP_APX_PLM_Santosh~~~skola2~~~2019-09-30 14:39:26@@@MF_PKMS_DA_TransferPickInboundPreProcessor.msgflow***MF_PKMS_DA_TransferPickInboundPreProcessor.msgflow' FROM DUAL;

Query:

SELECT SUBSTR( value, 1, first_position - 1 ) AS first_substr,
       SUBSTR( value, first_position + 3, second_position - first_position - 3 ) AS second_substr
FROM   (
  SELECT value,
         INSTR( value, '^^^', 1, 1 ) AS first_position,
         INSTR( value, '^^^', 1, 2 ) AS second_position
  FROM   test_data
)

Output:

FIRST_SUBSTR                                                                                                                                                               | SECOND_SUBSTR                                                                                               
:------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | :-----------------------------------------------------------------------------------------------------------
APP_APX_PLM_PostCategory~~~pavan anand~~~2019-09-26 15:03:39@@@MF_APX_PLM_PostBuyProgram.msgflow***MF_APX_PLM_PostBuyProgram1.msgflow***MF_APX_PLM_PostBuyProgram2.msgflow | APP_APX_PLM_Promo~~~skola2<br>~~~2019-09-30 14:34:11@@@MF_APX_PLM_Promo1.msgflow***MF_APX_PLM_Promo2.msgflow

db<>fiddle here

Use REGEXP_SUBSTR

https://dbfiddle.uk/?rdbms=oracle_11.2&fiddle=0981e7232b873934c5d4e833c141f47a

with tbl as 
(select 'APP_APX_PLM_PostCategory~~~pavan anand~~~2019-09-26 15:03:39@@@MF_APX_PLM_PostBuyProgram.msgflow***MF_APX_PLM_PostBuyProgram1.msgflow***MF_APX_PLM_PostBuyProgram2.msgflow^^^APP_APX_PLM_Promo~~~skola2~~~2019-09-30 14:34:11@@@MF_APX_PLM_Promo1.msgflow***MF_APX_PLM_Promo2.msgflow^^^APP_APX_PLM_Santosh~~~skola2~~~2019-09-30 14:39:26@@@MF_PKMS_DA_TransferPickInboundPreProcessor.msgflow***MF_PKMS_DA_TransferPickInboundPreProcessor.msgflow' str 
from dual)
select TRIM( '^' FROM REGEXP_SUBSTR(str, '.+?\^{3}')),
TRIM( '^' FROM REGEXP_SUBSTR(str, '\^{3}.+?\^{3}',1,1,'n'))  from tbl

output

APP_APX_PLM_PostCategory~~~pavan anand~~~2019-09-26 15:03:39@@@MF_APX_PLM_PostBuyProgram.msgflow***MF_APX_PLM_PostBuyProgram1.msgflow***MF_APX_PLM_PostBuyProgram2.msgflow  APP_APX_PLM_Promo~~~skola2~~~2019-09-30 14:34:11@@@MF_APX_PLM_Promo1.msgflow***MF_APX_PLM_Promo2.msgflow

I'm going to approach it a different way, and assume you are preparing to parse this multi-delimited string. Since the first level of delimiting seems to be by '^^^', let's use connect by to traverse the string and split out those elements, returning just the first 2 since that is your requirement. This returns the first 2 elements delimited by '^^^':

with tbl(str) as (
  select 'APP_APX_PLM_PostCategory~~~pavan anand~~~2019-09-26 15:03:39@@@MF_APX_PLM_PostBuyProgram.msgflow***MF_APX_PLM_PostBuyProgram1.msgflow***MF_APX_PLM_PostBuyProgram2.msgflow^^^APP_APX_PLM_Promo~~~skola2~~~2019-09-30 14:34:11@@@MF_APX_PLM_Promo1.msgflow***MF_APX_PLM_Promo2.msgflow^^^APP_APX_PLM_Santosh~~~skola2~~~2019-09-30 14:39:26@@@MF_PKMS_DA_TransferPickInboundPreProcessor.msgflow***MF_PKMS_DA_TransferPickInboundPreProcessor.msgflow' str 
  from dual
)
select regexp_substr(str, '(.*?)(\^\^\^|$)', 1, level, NULL, 1) element
from tbl
connect by level <= 2; --regexp_count(str, '\^\^\^')+1;

Note if you only want a particular element, replace "level" in the regexp_substr call with the number of the element.

Just copy your string when window pops up or replace with :str

select 
substr(:str,
instr(:str,'^^^',1,1) +3,
instr(:str,'^^^',1,2) - instr(:str,'^^^',1,1) - 3)
from dual;