"After eating his dinner, Tom turned on the TV".
"After missing the bus, we decided to drive into town".
"After finishing the cake, we started on the cheese".
The BBC website is poorly constructed and must be very confusing to learners. Nevertheless, I suspect this is all about a divided opinion on which category (part of speech) "after" belongs in. Traditional grammar, like that taught on the BBC grammar website, sees "after" as a preposition when it has a noun complement (object), but a subordinating conjunction when it introduces a clause.
But this view fails to recognise that in the very latest grammars, "after" is seen as a preposition which can take a noun or a subordinate clause as complement. And crucially, of course, prototypical preps take nouns as complement, so -ing verbs, when heading clausal complements to a prep (i.e. where nouns normally operate), are seen as having a similar nominal function and hence called gerunds.
The consequence of all this is that if, like the BBC, you take "after" as a subordinator, then an expression like "After eating his dinner" is a present participle clause. It's not a complement to anything, but simply an optional adjunct.
If, on the other hand, you take "after" to be a preposition, then "After eating his dinner" is a preposition phrase in which "eating his dinner" is a gerundial clause (with the gerund "eating" as head) functioning as complement to "after". In other words, it's a clause within a phrase, which is a common occurrence in English grammar.
Here's a link to 'EnglishClub' website which conveniently cites "after" as an example of a preposition, and states that (no exception) any verb that follows a preposition must be a gerund. They call the complement a phrase, whereas it's better to call it a clause: