In Russian, the accusative singular form (the direct object form) of animate masculine nouns whose stem ends with a consonant (and мальчик is one) coinsides with their genitive singular form: ведет что? - урок, but ведет кого? - мальчика (What is she teaching? - A class, but Who is she leading? A boy)
Она везёт мальчика домой. The infinitive of везёт is везти - to drive smb, to carry smb or sth by means of transport; мне везёт = I am lucky. If the action is performed on a regular basis, the verb возить is used, e.g.: Отец возит мальчика в школу каждый день = The father drives the boy to school every day. The relation between везти and возить is the same as the one between идти and ходить, нести and носить, вести and водить etc. There are 17 or so such pairs among Russian verbs of movement.
Only везти is used for driving people. "Вести (ведёт)" has lots of meanings, not just leading. It also means "to drive" (a vehicle), "to run" (a business), "to conduct" (negotiations), "to steer" or "sail" (a boat), "to keep" (a diary), and "вести себя" means "to behave". "Она приведёт мальчика домой" means "She will bring the boy home (and they will come on foot)"
I listened to the recording again, but the second syllable of она is not stressed. In fact, it is pronounced so closely with the first syllable of ведёт that it almost sounds like он велёт. I understand that any language, including English, undergoes pronunciation changes when spoken at normal conversatonal speed, but that does make it hard for those of us who are learning sounds and spelling.