Why does LEN() function badly underestimate cardinality in SQL Server 2014?
For the legacy CE, I see the estimate is for 3.16228 % of the rows – and that is a &quot;magic number&quot; heuristic used for column = literal predicates (there are other heuristics based on predicate construction – but the LEN wrapped around the column for the legacy CE results matches this guess-framework).  You can see examples of this on a post on Selectivity Guesses in absence of Statistics by Joe Sack, and Constant-Constant Comparison Estimation by Ian Jose.
-- Legacy CE: 31622.8 rows
SELECT  COUNT(*)
FROM    #customers
WHERE   LEN(cust_nbr) = 6
OPTION  ( QUERYTRACEON 9481); -- Legacy CE
GO

Now as for the new CE behavior, it looks like this is now visible to the optimizer (which means we can use statistics).  I went through the exercise of looking at the calculator output below, and you can look at the associated auto-generation of stats as a pointer:
-- New CE: 1.00007 rows
SELECT  COUNT(*)
FROM    #customers
WHERE   LEN(cust_nbr) = 6
OPTION  ( QUERYTRACEON 2312 ); -- New CE
GO

-- View New CE behavior with 2363 (for supported option use XEvents)
SELECT  COUNT(*)
FROM    #customers
WHERE   LEN(cust_nbr) = 6
OPTION  (QUERYTRACEON 2312, QUERYTRACEON 2363, QUERYTRACEON 3604, RECOMPILE); -- New CE
GO

/*
Loaded histogram for column QCOL:
[tempdb].[dbo].[#customers].cust_nbr from stats with id 2
Using ambient cardinality 1e+006 to combine distinct counts:
  999927
 
Combined distinct count: 999927
Selectivity: 1.00007e-006
Stats collection generated:
  CStCollFilter(ID=2, CARD=1.00007)
      CStCollBaseTable(ID=1, CARD=1e+006 TBL: #customers)
 
End selectivity computation
*/
 
EXEC tempdb..sp_helpstats '#customers';


--Check out AVG_RANGE_ROWS values (for example - plenty of ~ 1)
DBCC SHOW_STATISTICS('tempdb..#customers', '_WA_Sys_00000001_B0368087');
--That's my Stats name yours is subject to change

Unfortunately the logic relies on an estimate of the number of distinct values, which is not adjusted for the effect of the LEN function.
Possible workaround
You can get a trie-based estimate under both CE models by rewriting the LEN as a LIKE:
SELECT COUNT_BIG(*)
FROM #customers AS C
WHERE C.cust_nbr LIKE REPLICATE('_', 6);



Information on Trace Flags used:

2363: shows a lot of information, including statistics being loaded.
3604: prints the output of DBCC commands to the messages tab.