An explicit explanation.

Let's draw numbers on the balls to track them, shall we?

Box 1, Gold 1 = #1
Box 1, Gold 2 = #2
Box 2, Gold = #3
Box 2, Silver = #4
Box 3, Silver 1 = #5
Box 3, Silver 2 = #6

You choose a box at random. This is 1/3 probability. You choose a ball at random. This is 1/2 probability (within that box).

Thus, you have a 1/6 probability of pulling any specific ball here.

HOWEVER. The rules state YOU PULLED A GOLD BALL. Yes? Guaranteed, you are holding either ball #1, #2, or #3 in your hand. AND they all had equal probability of occurring. Thus, 1 out of 3 odds each.

So the odds you are holding each ball right now is:

#1 = 1/3
#2 = 1/3
#3 = 1/3
#4 = 0 (we did not pull a silver ball)
#5 = 0 (we did not pull a silver ball)
#6 = 0 (we did not pull a silver ball)

Everyone follow so far?

Let's look at what happens in each of the three cases. GIVEN that you know you pulled a gold ball:

1/3 times you Pulled #1: Your next pull is guaranteed to be #2, also a gold ball. = 100% gold
1/3 times you Pulled #2: Your next pull is guaranteed to be #1, also a gold ball. = 100% gold
1/3 times you Pulled #3: Your next pull is guaranteed to be #4, a silver ball. = 0% gold

Let's add these up.

(1/3 * 100%) + (1/3 * 100%) + (1/3 * 0%) = (1/3 * 1.0) + (1/3 * 1.0) + (1/3 * 0.0) = 2/3