Translation:The child is clean.
He meant that some consonants become aspirated, when they are followed by ㅎ. Like for example 구급함 (first aid kit) is pronunced like [구그팜], because ㅂ and ㅎ merge into ㅍ. 닦하다 (to clean) sounds like [다카다], because ㅎ adds aspiration to ㄲ.
Now in 깨끗하다 the case is a bit more complicated. Because ㅅ is followed by a consonant, the pronunciation is [깨끋하다]. But, since it is not just any random consonant, but ㅎ, the ㅅ gets additionally aspirated after changing into ㄷ. So the final pronunciation is [깨끄타다]. Long story short, ㅅ actually changes the way it sounds twice in a single word: ㅅ > ㄷ > ㅌ.
Is the the audio for the double consonant 깨 wrong, or...? When I looked up the pronunication of ㄲ - everything I've seen suggests that it's more aspirated and a slightly higher tone, but not actually repeated like this. Is this word an exception? Or is there another rule I don't know about?
I studied linguistics in college, and encountered several Korean experts who all vehemently disagreed about so-called tense consonants. Best explanation is the glottis is tensed up, causing a somewhat strained voicing onset. This can manifest as a pitch change or longer pause before voicing, but not always.
The ㅅ is a base consonant, and when a base consonant comes at the end of a syllable block, it changes sound to a weakened or un-pronounced ㄷ, sounding like the t in the American pronounciation of "button". It is a sound change rule, all base consonants fall under this rule. Watch Learn Korean with GO! Billy Korean on youtube. He has many informative videos about Korean and sound change rules.