a duce which means to bring, to come, to carry based on how it's used.
Here it is used transitively. Let's take a simpler example using the present: Eu mă duc la școală. (I go to school). Note that compared to Eu merg la școală (I walk to school), we have that mă because the verb acts on me. I can also say Eu te duc la școală (like a parent to a child saying I'll get you to school) so now we have te because it acts on you.
We can use it in a non-transitive manner to say Eu duc câinele acasa (I bring the dog home) so now it acts on the dog.
Transitivity refers to whether or not a verb can take a direct object. Your last example also uses "a duce" in a transitive manner.
What is used here is actually a reflexive verb (i.e. a transitive verb whose direct object is the same as the subject). It's very confusing, because Romanian reflexive pronouns are the same as object pronouns (except in third person), but the idea is that these two sentences are different:
Eu mă iubesc. (reflexive)
Ea mă iubește. (not reflexive, but transitive)
which is more clear in English:
I love myself.
She loves me.
So, if a reflexive verb is just a special case of a transitive verb why does it matter? This can be seen using third person pronouns:
El îl duce la școală.
El se duce la școală.
In both cases, the verb is transitive, but only the second sentence has the same subject and object.