Supposing I am a speaker on TV and going to tell the viewers a method of something e.g. a particular dress. I'll demonstrate it by wearing. Can I say, 'Let's now teach you how to wear this.' or Should I say, "Let me teach you how to wear this."? I mean when only the speaker will be involved in doing this, is it right to use "let's+ verb (first form)"? Isn't let's used when the speaker and the listener will both be involved in doing what has just been said or suggested?
You are correct; let's is a contraction of let us and so involves both the speaker and the listener.
Your verison of the phrase:
Let me teach you how to wear this
Is grammatically correct and would not be misunderstood.
However, in the context of television - the presenter will commonly try to make the audience feel involved in what they are doing, even if the actions are only being performed by the presenter.
This means that at times, the presenter may use slightly odd constructs unintentionally to try to include the audience.
The phrase you provided falls into this area.
Let us now teach you how to wear this dress
Switches between talking about the presenter and audience as a group (let us), and then talking about the audience as a separate entity (teach you). As such, it is understandable, but does come across slightly odd - and is perhaps less technically correct.
A more correct version of the presenter's speech could be:
Let us now see how we can wear this dress.
In this case, both the audience and presenter are seeing how the dress can be worn together - it just happens the presenter is the one wearing the dress.
Importantly, this version is consistent in treating the audience and presenter as a single group (let us with we).
To be clear though, both phrases would not be surprising to hear on television. A perfect sentence is less important to television, than making the audience feel engaged and involved - in most cases.