They're not tests of intelligence so much as tests of how similar your pattern matching is to that of the test authors.
Well... they are tests of intelligence. A valid test of intelligence is one that lets you determine ("predict") the intelligence of a testee from their answers. For that, you need intelligent and unintelligent people to have measurably different answer patterns, but there is no requirement that the intelligent answer patterns be objectively correct. You could have a test that was entirely subjective ("Who was the better artist, Michelangelo or Jackson Pollock?") and still use it to predict intelligence[1].
[1] That's an invented example; I'm not claiming that the particular question of Michelangelo vs. Jackson Pollock is actually predictive of intelligence, just pointing out that it could be.
A better test suite would likely provide an oracle capable of generating new images belonging to the left or right and asking the program to judge which side it belongs to.
The rule is that the shapes on the left all have exactly three sides as single straight lines. (Similarly, the shapes on the right have exactly three "unusual" sides.)
That said, though I've referred to "the rule" and the post author also refers to "the solution" to a Bongard problem, they don't and can't have authoritative solutions, just as sequence completion problems can't. When we talk about "correct" answers to this style of question, what we mean is "what tends to occur to smart people".