No, it's not. "Il" is singular and means "he", "leur" is plural and means "their". So the combination "il" + "leur" is not possible; only the combination "ils" + "leur" is. In this sentence "leur" becomes "leurs" with an "s" at the end, because it changes with the noun: chaussures is plural.
It makes sense if he's putting their shoes on. Like a dad getting his kids ready to go outside ...
Either by reading "ils mettent" or listening ("il met" vs "ils mettent"), this can only be understood as "they". The "t" in "il met" is silent. You can listen to the audio and hear the "t" quite clearly in "ils meTtent" which is a clear audible clue that it is the third person plural conjugation of the verb.
The only explanation I could give for this is that the 't' on 'met' is silent, unlike the audio for this question.
Cannot see why the verb "mettre" cannot be translated as "to put", not "to put on". People can simply put their shoes somewhere, right?
They can, yes, but then I think you have to change the verb to something like poser, because it's my understanding that mettre + [un vêtement] means "to put on [an item of clothing]."
think: les étudiants aiment leur professeur. ( = their language teacher); Les étudiants aiment leurs professeurs. (= their teachers of history, maths etc.). Les parents aiment leur fils/ ...leurs enfants. ( only one son/ two or more children)
I think because mettre is 'to put' not 'to wear' and putting on shoes is different than wearing them (in other words, they are in the act of putting on shoes)
There is a tiny bit of overlap with the English "wear". If you look in your closet and ask yourself, what shall I wear today, the French verb would be "mettre" (meaning "wear" in the sense of I am going to put it on). However, once you have put on the clothing, you are now wearing it. This is the verb "porter". In short, when "mettre" is used in the context of clothing, you are safe in thinking that it means "to put on". When "porter" is used in the context of clothing, you are safe in thinking that it means "to wear".
For anyone confused
leurs means "their" indicating plurality of the situation
Therefore Il met ---becomes--- Ils mettent
I think that's actually backwards. I think the hard "t" in "meTTent" tells you that it is "mettent," rather than "met" and therefore "ils," not "il." Then logically it has to be "chaussures," not "chaussure," because if between them they own a single shoe they cannot both or all but it on at the same time, and if "chaussures," then it must be "leurs." I am struggling with French so would appreciate if a native speaker would confirm if this is correct or not.