I appreciate David Bowman's answer above, with one exception: he writes, "Isn't he/him the subject of 'speak'? No. If that were the case, we would use 'speaks,' which is the third person singular form of 'speak.'" Actually, the "him" is the subject of "speak". I don't know the terminology in English, but it is akin to the accusative subject of an infinitive in Latin (if I remember correctly). It is very common for a direct object to serve also as the subject of a following verb in its barest form (the "infinitive" in other languages, though usually without the "to" in English): "Watch me do this": the "me" is the object of "watch" and the subject of "to do" (for I am the one 'doing', no?); "Help me (to) do this"; "Let me do this", etc.

I appreciate David Bowman's answer above, with one exception: he writes,

Isn't he/him the subject of 'speak'? No. If that were the case, we would use 'speaks,' which is the third person singular form of 'speak.

Actually, the "him" is the subject of "speak". I don't know the terminology in English, but it is akin to the accusative subject of an infinitive in Latin (if I remember correctly). 

It is very common for a direct object to serve also as the subject of a following verb in its barest form (the "infinitive" in other languages, though usually without the "to" in English):

Watch me do this
Help me (to) do this
Let me do this; etc.

The "me" is the object of "watch", "help", and "let"; and the subject of "to do" (for I am the one 'doing', no?).