Like in Russian the verb "hear" requires the accusative case and therefore "he" needs to be inflected and becomes "him". - The people heard HIM (do something).
Now, you can only use "laughS" in connection with the the personal pronouns "he", "she" and "it". Therefore, you can't say "they heard him laughs".
I'm sure there must be a grammar rule which I don't know, unfortunately.
And yes, you could say "they heard him laughing". But there's a subtle difference in meaning. They heard him laughing means, he had been laughing for a while. They heard him laugh means he either laughed shortly in one situation or he laughed in different situations but never anyone around him joined in.