You can say "he is sick and weak." But you want to emphasize the fact that he is BOTH sick and weak. So you use the structure "both..and" to convey this meaning.
In Esperanto, as many languages, this kind of structures (both..and, either..or, neither..nor, etc) is made by repeating the conjunction:
- He is both sick and weak = Li estas kaj malsana kaj malforta.
- He is either sick or weak = Li estas aŭ malsana aŭ malforta.
- He is neither sick nor weak = Li estas nek malsana nek malforta. (usually, "nek" is used in pairs.)
You may want to know more. Take a look at this chapter in PMEG.
This would have confused the hell out of me ( as a native English speaker) if I had not read the tips and notes section on the website. Those who only use the smartphone app should consider reviewing the "tips and notes" that can be accessed through the website version of Duolingo. I was prepared to see this because of the discussion of paired conjunctions.