"Today, you cannot put on that red tie."
The construction at play here is ～てはいけません (-te wa ikemasen), which means "you're not allowed to ~" or "you can't ~". To tell someone they are not allowed to do something, the thing that you are not allowed to do is put in the -te form (in this case ネクタイを締めて nekutai o shimete), then add はいけません (wa ikemasen).
Nekutai o shimete wa ikemasen.
You can't wear a tie.
I can't explain the grammatical reasons for why it's the -te form, and you might be onto something, but I think for most people it's clearest and simplest to just remember:
～てはいけません = not allowed to ~
I can explain the selection of the verb, at least. Wearing/putting on different things uses different verbs in Japanese. 締める (しめる), which means "to fasten/tighten" is used to talk about wearing/putting on belts and neckties, since these things are fastened around one's body. 閉める (しめる) means "to close/shut" and is a homophone and likely a polyseme of 締める (しめる).
This is the first time I encounter double particle は, both here and in the wild. I do understand that the second particle acts as an nominalizer and perhaps can be replaced with some other combinations of particles, like we can see in another exercises. Is it OK to use multiple topic marking particles in one sentence without being frowned upon?