Translation:Our professor uses his tongue like a sword.
Ah. I haven't looked at the notes here. Kiel is basically like the word for "how" in a lot of European languages. It can be used for how, but it is also used for "like" and "as", comparing things. In English, we can use "how" before a clause but not before a single noun phrase. Thinking of the full sentence might help you.
Kiel vi iras hejmen? = How do you get home?
Mi laboras kiel instruisto (laboras). = I work as a teacher (works). (= I work how a teacher works.)
Nia profesoro uzas sian langon kiel (li uzus) glavon. = Our professor uses his tongue like (he would use) a sword. (= Our professor uses his tongue how he would use a sword.)
Nia profesoro uzas sian langon kiel glavo (uzas sian langon). = Our professor uses his tongue like a sword (uses its tongue). (!?!?!) (= Our professor uses his tongue how a sword would use its tongue.)
Mi mortigos vin kiel hundo (mortigus vin) = I will kill you like a dog (would kill you)
(= I will kill you how a dog would kill you.)
Mi mortigos vin kiel (mi mortigus) hundon. = I will kill you like (I would kill) a dog. (= I will kill you how I would kill a dog.)
It's the same as sentences with "ol":
Mi amas vin pli ol (mi amas) lin. = I love you more than (I love) him.
Mi amas vin pli ol li (amas vin) = I love you more than he loves you.
So, in this part of the lesson, my first question was to identify the Esperanto word for "glove", which I recalled from past lessons was "ganto." (I've just confirmed this using Duo's dictionary.) But the picture of the glove (actually a mitten, but I won't split that hair here), had the label "glavo." Then I got this translation, which at first puzzled me - our professor uses his/her tongue as a glove? So I hovered and found that glavo means sword. Now how do I go back to the picture of the glove and report that it's not a glavo/sword?
If two things are joined by "kiel" meaning "like", then those two things must be in the same case (otherwise they cannot be joined grammatically): "langon" is in the objective case because it is the object of the transitive verb "uzas" and "glavon" is in the objective case because it is joined to "langon".
As a conjunction "kiel" can join two things far apart in a sentence. For example, La lango de la profesoro estas akra kiel glavo. (Kiel glavo, la lango de la profesoro estas akra. Lia lango kiel glavo estas akra.)