It is used correctly, the sentence isn't awkward although it is very long. Where in this case is used to demonstrate/explain in more detail how the author expresses an optimistic attitude.
In Shane Koyzcan’s poem “Instructions for a Bad Day,” the author expresses an optimistic attitude towards life where he encourages us to forget the burden of the past
can be written as:
In Shane Koyzcan’s poem “Instructions for a Bad Day,” the author expresses an optimistic attitude towards life. He does this by encouraging us to forget the burden of the past
A shorter example would be:
The man talked about different cats, where he discussed their colour and size.
He talks about different cats, and in more detail, he discusses their colour and size.
To answer your other question:
I understand the author‘s worry, confusion, and question about the purpose of life, to which I have the same feeling.
You have used to which correctly. However I would suggest making question plural (questions) because it is not a specific single question, but an attitude that encompasses a range of questions. I would also change feeling to feelings for the same reason, that the author has many feelings about the situation, rather than one specific feeling.