"Please wear a shirt, and then put on a tie."
Though I can not to explain grammatically.
This sentence becomes to make from two sentences as the following.
Please wear a shirt. And then please put on a tie.
That is the same meaning.
This is way that two sentences connect to be one sentence.
I think it's "wear, and..." Present progressive "te" on a verb can indicate that the sentence is not over, another verb is coming. In this case "and put on a necktie".
Just adding to this, I think the use of て-form verbs indicates a sequence of events being combined into a single sentence (which is what Sora was trying to get at, I think).
To give another example, if I was explaining my morning routine to someone:
It sounds very stilted to say "I wake up at 6:30. I eat breakfast. I put on my uniform. I go to school." To squish them all into one sentence (also sounds a bit odd, like a 5 year old saying "and then... and then...", but bear with me):
It translates more like "I wake up at 6:30, then eat breakfast and put on my uniform, then go to school."
Importantly, the sequence and the tense is maintained throughout, determined by the tense of the last verb.
Kite refers to putting on the shirt. Shimete refers to the tie and other accessories i think.
しめて doesn't universally refer to all accessories, only those that you "tighten" or "enclose" around something, e.g. ties and belts.
In the case of this sentence, しめて is the て-form of the verb "to tie, to fasten", with the て-form being indicative of (again in this case) a request.
Depending on the context and the kanji, しめて could also be the て-form of one of these verbs:
- "to strangle" 【絞めて】
- "to comprise, to account for" 【占めて】
- "to close, to shut" 【閉めて】
Google returns nearly 400000 pages with "ネクタイを着" indicating that 着る is a common verb to use with ネクタイ, yet Duolingo only accepts 締める (which gets less hits).