"Widely" is an adverb that means travel over an extensive area.
"Wide" is also an adverb that gives information about the extent of the phrase
"Width" is a noun and not appropriate, and "widen" is a verb "to widen."
That is why "widely" and "wide" are the correct answers.
They are both adverbs, and they both give appropriate extra information.
Which way does Mr. Smith like traveling around the world? Widely.
In what way did her eyes open (extent) when she heard the good news? Wide.