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I think it's the way it's written that causes confusion.

Roll die one: 1/6 chance of being a 6.

At this point, the chances of 6-6 are indeed the same as the chances of 6-5.

The article isn't however exploring that scenario. It's treating 6-5 as meaning either of the dice shows the 6, and the other shows the 5.

So 6-6 needs die one to be a 6, and die two to be a 6.

6-5 needs die one to be a 6, two to be a 5

5-6 needs one to be a 5, two to be a 6

Thus the combination of 6 and 5 has two ways in which it can be reached, which is twice as many as the ways of rolling two 6s.