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Consider the following simple MCVE:

SET STATISTICS IO, TIME OFF;
USE tempdb;

IF OBJECT_ID(N'tempdb..#t1', N'U') IS NOT NULL DROP TABLE #t1;
CREATE TABLE #t1
(
 r int NOT NULL
);

IF OBJECT_ID(N'tempdb..##t1', N'U') IS NOT NULL DROP TABLE ##t1;
CREATE TABLE ##t1
(
 r int NOT NULL
);

IF OBJECT_ID(N'dbo.s1', N'U') IS NOT NULL DROP TABLE dbo.s1;
CREATE TABLE dbo.s1 
(
 r int NOT NULL
 PRIMARY KEY CLUSTERED
);

INSERT INTO dbo.s1 (r)
SELECT TOP(10000) ROW_NUMBER() OVER (ORDER BY (SELECT NULL))
FROM sys.syscolumns sc1
 CROSS JOIN sys.syscolumns sc2;
GO

When I run the following inserts, inserting into #t1 shows no stats I/O for the temp table. However, inserting into ##t1 does show stats I/O for the temp table.

SET STATISTICS IO, TIME ON;
GO

INSERT INTO #t1 (r)
SELECT r
FROM dbo.s1;

The stats output:

SQL Server parse and compile time: 
 CPU time = 0 ms, elapsed time = 1 ms.
Table 's1'. Scan count 1, logical reads 19, physical reads 0, read-ahead reads 0, lob logical reads 0, lob physical reads 0, lob read-ahead reads 0.

 SQL Server Execution Times:
 CPU time = 16 ms, elapsed time = 9 ms.

(10000 rows affected)

INSERT INTO ##t1 (r)
SELECT r
FROM dbo.s1;

SQL Server parse and compile time: 
 CPU time = 0 ms, elapsed time = 1 ms.
Table '##t1'. Scan count 0, logical reads 10016, physical reads 0, read-ahead reads 0, lob logical reads 0, lob physical reads 0, lob read-ahead reads 0.
Table 's1'. Scan count 1, logical reads 19, physical reads 0, read-ahead reads 0, lob logical reads 0, lob physical reads 0, lob read-ahead reads 0.

 SQL Server Execution Times:
 CPU time = 47 ms, elapsed time = 45 ms.

(10000 rows affected)

Why are there so many reads on the ##temp table when I'm only inserting into it?

Minimal logging is not being used when using INSERT INTO and global temp tables

Inserting one million rows in a global temp table by using INSERT INTO

INSERT INTO ##t1 (r)
SELECT top(1000000) s1.r
FROM dbo.s1
CROSS APPLY dbo.s1 S2;

When running SELECT * FROM fn_dblog(NULL, NULL) while the above query is executing, ~1M rows are returned.

One LOP_INSERT_ROW operation for each row + other
log data.

The same insert on a local temp table

INSERT INTO #t1 (r)
SELECT top(1000000) s1.r
FROM dbo.s1
CROSS APPLY dbo.s1 S2;

Only going up to 700 rows returned by SELECT * FROM fn_dblog(NULL, NULL)

Minimal logging

Inserting one million rows in a global temp table by using SELECT INTO

SELECT top(1000000) s1.r
INTO ##t2
FROM dbo.s1
CROSS APPLY dbo.s1 S2;

SELECT INTO a global temp table with 10k records

SELECT s1.r
INTO ##t2
FROM dbo.s1;

Time and IO Statistics

SQL Server parse and compile time: 
 CPU time = 0 ms, elapsed time = 0 ms.
Table 's1'. Scan count 1, logical reads 19, physical reads 0, read-ahead reads 0, lob logical reads 0, lob physical reads 0, lob read-ahead reads 0.

 SQL Server Execution Times:
 CPU time = 16 ms, elapsed time = 10 ms.
SQL Server parse and compile time: 
 CPU time = 0 ms, elapsed time = 0 ms.

Based on this blogpost we can add TABLOCK to initiate minimal logging on a heap table

INSERT INTO ##t1 WITH(TABLOCK) (r)
SELECT s1.r
FROM dbo.s1

Low logical reads

Table 's1'. Scan count 1, logical reads 19, physical reads 0, read-ahead reads 0, lob logical reads 0, lob physical reads 0, lob read-ahead reads 0.

(10000 rows affected)

Part of an answer by @PaulWhite on how to achieve minimal logging on temporary tables

No. Local temporary tables (#temp) are private to the creating
session, so a table lock hint is not required. A table lock hint would
be required for a global temporary table (##temp) or a regular table
(dbo.temp) created in tempdb, because these can be accessed from
multiple sessions.

Creating a regular table to test this:

CREATE TABLE dbo.bla
(
 r int NOT NULL 
);

Filling it up with 1M records

INSERT INTO bla 
SELECT top(1000000)s1.r
FROM dbo.s1
CROSS APPLY dbo.s1 S2;

>1M logical reads on this table

Table 's1'. Scan count 17, logical reads 155, physical reads 0, read-ahead reads 0, lob logical reads 0, lob physical reads 0, lob read-ahead reads 0.
Table 'bla'. Scan count 0, logical reads 1001607, physical reads 0, read-ahead reads 0, lob logical reads 0, lob physical reads 0, lob read-ahead reads 0.
Table 'Worktable'. Scan count 0, logical reads 0, physical reads 0, read-ahead reads 0, lob logical reads 0, lob physical reads 0, lob read-ahead reads 0.

Paul White's answer explaining the logical reads reported on the global temp table

Generally, logical reads are reported for the target table when the
insert is not minimally logged.

These logical reads are associated with finding a place in the
existing structure to add the new rows. Minimally-logged inserts use
the bulk-loading mechanism, which allocates whole new pages/extents
(and so does not need to read the target structure in the same way).

Conclusion

The conclusion being that the INSERT INTO is not able to use minimal logging, resulting in logging every inserted row individually in the log file of tempdb when used in combination with a global temp table / normal table.
Whereas the local temp table/ SELECT INTO/ INSERT INTO ... WITH(TABLOCK) is able to use minimal logging.