It sounded weird to me when I first heard about this usage too, in the context of a grammar lesson in middle school. Much like the linked page with the examples in the question does, the teacher taught us that using possessive pronouns (also known as genitives) is the only grammatical way to mark subjects of gerund clauses. And indeed, it is the the more traditional, formal way to do it, even though using object pronouns (accusatives) is also quite common.
This is discussed in the Cambridge Grammar of the English Language, in chapter 14 “Non-finite and verbless clauses”, section 4.3. The main thrust of this section is to lay waste to the traditional distinction between gerund and present participle clauses, and argue that they all belong to a single inflectional category, calling them all gerund-participles. However, there is a paragraph explaining use of genitives with gerunds:
There is one respect in which ‘gerund’ and ‘present participle’ clauses differ in their internal form: with ‘gerunds’ the subject may take genitive case, with plain or accusative case a less formal alternant, but with ‘present participles’ the genitive is impossible and pronouns with a nominative–accusative contrast appear in nomiative case, with accusative an alternant restricted to informal style. Compare then:
 i. She resented his/him/*he being invited to open the debate.
ii. We appointed Max, he/him/*his being much the best qualified of the candidates.
What this is saying is that the constructions that are traditionally known as gerunds (as in example 39i) can take either the genitive (his) or the accusative (him) as subject, noting that accusative is “a less formal alternant”. The nominative (he) is not possible as the subject of a gerund. They add that in participial clauses that can have a subject, there is a similar situation: both the nominative (he) and accusative (him) are possible, with the accusative being less formal, but the genitive (his) is not possible.
The link where the original poster’s examples come from exhibits the common problem in grammar advice of confusing informal style with incorrect grammar. The versions of the examples with accusative instead of genitive (e.g. What do you think about him buying such an expensive car?) are perfectly grammatical and simply just a less stuffy style than those with the genitive. If you want my advice, you will find many examples of gerunds with accusative subject even in formal academic writing, so you should feel free to use whichever of the two formulations that seems most natural to you.