"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.
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.
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】
You have to look into the sentences that google returns, or you need to surround your search text with double quotes. Google does an intelligent search so that it also returns pages with high relevence. If I use double quote "ネクタイを着" I only get 32 pages. These are more common in terms of search hits: ネクタイをする, ネクタイをしめる, ネクタイを結(むす）ぶ, ネクタイを着用（ちゃくよう）する
the real problem is that the English that they are giving us is unnatural. I can understand it but it really does not make sense. The English should be either 1) "Please wear a shirt and tie.' 2) "Please put on a shirt and tie." In the real world we don't wear a tie without already having put on a shirt unless you are drunk. So, the English sentence is stilted or weird in normal situations.