Sort of, technically, but it's not considered to be good Esperanto. (began is "komenciĝis") But if you forgot the proper word for "began", then you could certainly use "malĉesis" and most probably be understood.
Just a note: in this context it should be komenciĝi, for the rain itself starts and does not initiate something else. (Considering ĉesi is an intransitive verb this would make more sense.)
Oops, I forgot for a sec that "komenci" is transitive. Thanks, I've corrected my comment.
"The rain has stopped" and "The rain stopped" mean the same thing in English. If the question was marked wrong, try reporting the error.
They do not mean the same thing in English.
But they are both reasonable translations of the Esperanto sentence.
‘The rain has stopped’ implies that it still hasn't rained since, just like ‘I have lived here for 5 years’ implies that you are still living there, whereas ‘I lived here for 5 years’ indicates that you don't necessarily live there still. (But generally no one really bothers with this and I'm not sure that it is always interpreted like this; I just learned it like this at school years ago.)
The audio on this one is very bad, the enunciation of the words are totally lost on my Toshiba laptop speakers. I agree with "ActualGoat". I think that some of the audio in this course is so "flat" and "fast" that I wish that they had hired a Cuba or a South American Esperanto speaker for the audio. I find listening to South American or Cuban Esperanto speaker a joy to listen to. (As a newbie to the Esperanto Language, I [and I am sure others, too] need as many "signals" to listen to, and correctly identify the spoken words.)