Actually in french it can also mean to buy. My source: http://www.larousse.com/en/dictionaries/french-english/offrir/55389 . Note the phrase: je vous offre un café/un verre ? can I buy you coffee/a drink ?
i thought boire and manger per se meant 'to drink' or 'to eat' as that. As in j'aime boire et j'aime manger.
My wild guess is that some verbs require this à preposition in some constructions with the infinitive.
There's some extra voodoo going on here but I'm not sure what makes the doll so I'm not sure how to proceed! I hate dolls.
Actually, you're right. Some verbs do require the "à" after them. This page might help, as well as other pages within the series: http://french.about.com/od/grammar/a/preposition_a_2.htm
Lui means 'to him' and is an indirect object, whereas Le means 'him' and is a direct object. So if you said, 'Il l'offre ...', then you would be saying 'I buy him'. 'Il lui offre...' means 'I buy something for him'. It's a little difficult to compare to English because giving something and giving to something sound the same. Like I gave the dog could mean that you are giving the dog to someone or you are giving the dog something.
If you said "le offre", it would be "l'offre." I don't actually know whether this verb can be used this way (I imagine it can be), but I think the difference is that if you used le/la, it would be more like the person himself was being offered (or bought): "He offers HIM (to someone)" whereas with lui, you're offering something TO him.
Since we are given no context (does this happen in a restaurant or in a home or in a refugee camp?), Duo should accept either "buy" or "offer" in the English translation. If the context is a restaurant, "buy" is correct, but if it's in a home or other place that doesn't customarily charge for food, then it's "offer." Collins French-English Dictionary supports this with the following example sentence in the definition of the verb "offrir": "Elle lui a offert à boire. She offered him a drink."