Here's the weird thing about "it's" as opposed to the usual apostrophe and "S" to denote ownership (e.g., "Stephen's"). For whatever reason, when you denote ownership using the pronoun "it," you drop the apostrophe: "its."
An easy way to determine whether an apostrophe should be present in "its" is to read "it's" as "it is." So in this case, putting "it's" between "fried" and "circuits" would lead to "The robot fried it is circuits." Clearly wrong, so we can rule out the last two options listed.
And "The robot fried it circuits" wouldn't work either, since "it" by itself does not specify a relationship between the robot and the circuits. There goes the second option.
That leaves us with the first option. Let's double check to make sure that's the right one by expanding the contractions where applicable: "...that is why it is not working." Perfect.
