Answer by Professional Japanese Teacher
This phrase is used when answering the phone. You may be wondering why it is used. When the telephone was first introduced in Japan in 1890, calls could only be made between two parties with the help of telephone operators. When telephone operators started talking, they said “します します mooshimasu mooshimasu” which means “I will start talking”.
The word します is actually the humble form of います iimasu (to talk). The humble form is used so that the telephone operator will not sound rude to the person making the call. It was later shortened to moshi moshi.
As to why the word is repeated, some say it was because the connection was poor then so telephone operators usually repeated the word to ensure that the callers could hear them.
Take note that moshi moshi may sound a bit casual and is therefore not used for business calls.