Yes it does.  With the left side,

 1 + (2 + 3) 

the parentheses are around the 2 and the 3.
That means you are ASSOCIATING the 2 and 3 together. 
Then you have

1 + 5

which gives 6. 

Now with the right side,

(1 + 2) + 3

the parentheses are around the 1 and the 2.
That means you are ASSOCIATING them together. 
Then you have

3 + 3

which also gives 6.

So the associative property says that if you have
three numbers to add, it doesn't matter whether you
associate the first two, and them, and then add the 
third number, or whether you associate the last two,
add them, then add the first number to what you got,
you will always get the same final answer.

That principle also works for multiplying three numbers

2 x (3 x 4) = 2 x 12 = 24

(2 x 3) x 4 = 6 x 4 = 24

So 2 x (3 x 4) = (2 x 3) x 4

But it DOES NOT work for subtraction 

9 - (6 - 2) = 9 - 4 = 5

But

(9 - 6) - 2 = 3 - 2 = 1

So there is NO associative property for subtraction.

It also DOES NOT work for division.

So we say ADDITION and MULTIPLICATION both have the ASSOCIATIVE
PROPERTY, but SUBTRACTION and DIVISION do not.

Edwin