Well, it's the order of operations..
SELECT
t1.*
FROM
table1 t1
INNER JOIN table2 t2
LEFT JOIN table3 t3 ON t3.table3_ID = t2.table2_ID
ON t2.table2_ID = t1.table1_ID
could be rewritten as:
SELECT
t1.*
FROM
table1 t1 -- inner join t1
INNER JOIN
(table2 t2 LEFT JOIN table3 t3 ON t3.table3_ID = t2.table2_ID) -- with this
ON t2.table2_ID = t1.table1_ID -- on this condition
So basically, first you LEFT JOIN t2 with t3, based on the join condition: table3_ID = table2_ID, then you INNER JOIN t1 with t2 on table2_ID = table1_ID.
In your second example you first INNER JOIN t1 with t2, and then LEFT JOIN the resulting inner join with table t3 on the condition table2_ID = table1_ID.
SELECT
t1.*
FROM
table1 t1
INNER JOIN table2 t2 ON t2.table2_ID = t1.table1_ID
LEFT JOIN table3 t3 ON t3.table3_ID = t2.table2_ID
could be rewritten as:
SELECT
t1.*
FROM
(table1 t1 INNER JOIN table2 t2 ON t2.table2_ID = t1.table1_ID) -- first inner join
LEFT JOIN -- then left join
table3 t3 ON t3.table3_ID = t2.table2_ID -- the result with this
EDIT
I apologize. My first remark was wrong. The two queries will produce the same results but there may be a difference in performance as the first query may perform slower than the second query in some instances ( when table 1 contains only a subset of the elements in table 2) as the LEFT JOIN will be executed first - and only then intersected with table1. As opposed to the second query which allows the query optimizer to do it's job.