Sarah writes as neatly as her brother = Sarah and her brother both have the same level of neat handwriting.
James speaks as quickly as John = James and John both speak quickly.
The star was shining as brightly as all the others = the stars were of equal brightness.
Another use is when the second half of the sentence is "as + person + verb". This is when someone does/did something/something is/was done in the "most + adjective = as + adverb + as" way possible.
Please tidy your room as quickly as you can.
He walked to school as slowly as he could.
The boy sang as loudly as he could, but we still couldn't hear him!
Student or Learner