"Vous le remerciez."
Translation:You thank him.
"Le" has different functions and meanings in French.
You can use it attached with a noun -- Le chat, le père, le jeudi -- and we name it "article défini". You translate this one by "the" or by nothing accordind to the cases.
You can use it instead of a singular and masculine noun and then we name it "pronom personnel" (you don't have to remember the name of the function but it's to insist on the fact that if they look the same they are really different words ;p ). But (that's the tricky part), you can ONLY use it to complete a verb and ONLY if the noun which is replaced by "le" would be directly said after the verb (there is no "à", "de" or something else between them) and ONLY if we know which noun he is replacing (in the other case it's useless)
I will try to give you clear examples.
with the verb remercier = to thank. "Ton fils me fait un cadeau. Je remercie ton fils". We have two sentences, the first one means "your son gives me a gift", the second one is "I thank your son". In English you won't say that because it's annoying to have "your son" twice so you say "you son gives me a gift", "I thank him" because him can replace the noun "your son" and we know who him is referring to. In French that's the same, we use le (pronom personnel) instead of the second "ton fils" because "ton fils" is singular, masculine, directly put after the verb ("Je remercie ton fils" and not "Je remercie à ton fils" or "Je remercie de ton fils) and we know who "le" will be referring to. In this case the only difference is that when you use a "pronom" you put it before the verb and not after. So you say "Ton fils me fait un cadeau." "Je le remercie" (and not "Je remercie le" which is really a big mistake)
if I use "ta fille" or "tes fils" or "tes filles" you can't use "le" because the first one is feminine and the other one are plurial. (in reality you can use "la" or "les" and except for the gender and the numbers the rules are the same : "ta fille" - "je la remercie", "tes fils" or "tes filles" - "je les remercie")
it the "J'aime bien ton fils alors je parle à ton fils" you can't replace the second "ton fils" by "le" because there is "à" between "parle" and "ton fils" ( in reality there are other words to do that)
So I hope that now you understand why we use le in this case. Theoretically this sentence alone doesn't mean a lot because we don't know who "le" is referring to but it's an example and we can imagine that this sentence is taken from a conversation in which we can determine it.
If you are interested into the "pronoms", which are really common in French to avoid really long sentences, I can write a complete lesson about all the other cases ( what do we do when there is a "à" or an other word between the verb and the noun, what do we do if I don't want to refer to a noun but to me, you, we, how can we mix all these rules) if you want it ;)
No, "lui" is an indirect object pronoun. Here the "him" is a direct object pronoun.
Some examples of indirect object pronouns:
Je parle à Abigail. Je lui parle. Abigail is the indirect object here. I speak to her/a lui. (verb + preposition)
Je donne la lettre à Michel. Je lui donne la lettre. or Je la (la lettre) lui (Michel) donne.
The direct object is "la lettre" "la" (the item being given). The indirect object (Michel) receives the direct object.