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And how do I say it right?

This should be an explanation:

but I still don't understand that why does the sentence "I lonely walked around the park" does not make any sense

You actually can say that, but it sounds very poetic/old fashioned rather than like natural, everyday speech and you probably need some extra punctuation to clarify for modern speakers.

As the passage that you quoted explains, lonely is an adjective, not an adverb, so it can only describe a noun, not a verb (that is, it can describe a person, place, or thing, but can't describe an action). This means lonely cannot describe the way the person in your sentence is walking, but it could describe the person.

The problem with the sentence as-written is the placement of the word lonely. English usually puts adjectives in front of the noun they modify (red rose, beautiful dreamer, happy children, etc.), so without any other clues we are going to want to connect lonely with the word the word that follows it. In this case, that word is walked, but this doesn't work because it's not a noun.

We can "fix" this by interpreting lonely as an attributive clause. Attributive clauses modify the noun that immediately precedes them. Usually they're more than one word, but they don't have to be.

As I mentioned, this will sound poetical, and actually parallels some existing poetry:

When, in disgrace with fortune and men’s eyes,

I all alone beweep my outcast state

—William Shakespeare, Sonnet 29

Soon as the portals ope; I lonely tread

The sacred paths conductive thro' the choir

—Rev. Alsop, A Description of the Choir at St. Peter's Church, 1738

Thus wrapped in gloom I lonely walk

Life's dark and dreary way

—Joshua Ross, "My Ruling Star", 1855

As you can tell from the dates, it's also a very old-fashioned construction (there are other examples, but all that I've found are at least a hundred years old), so this won't be the first interpretation that most modern speakers will guess. We can make it a bit clearer by enclosing lonely in commas:

I, lonely, walked around the park.

However, if you want to have a more straightforward sentence that everyone will understand, you probably need to reword to something like

[Feeling] Lonely, I walked around the park.

or

I walked around the park, all alone.