"Toast", in English, means "a call to a gathering of people to raise their glasses and drink together in honor of a person or thing". The expression "to raise a toast" means when one person makes a short speech to invite everybody in the room to drink together to honor someone or something.
What we now know as DJ-ing, which was originally creating club music by manipulating phonograph records on a pair of turntables, originated in Jamaica. (the term "DJ" originaly meant "disk jockey" or somebody who plays records for a living.) At first the Jamaican dance-hall DJ-ing was just spinning records, but at some point a DJ got a microphone and began to talk to the audience while spinning records in order to get the audience excited about dancing and having a good time.
The story goes that while they were mixing the music, DJs got in the habit of using the microphone to call out the names of friends as they arrived in the club, and to encourage the audience to drink by "raising toasts". This quickly evolved into the DJs entertaining the crowd by rhyming vocal phrases in time to the rhythm over the phonograph records they were mixing, and this became known as "toasting". Originally it was only something done live in the club in the moment, as was also the case with the mixing of the spinning phonograph disks on the turntables, but at some point a record label decided on the idea of making new recordings that incorporated music taken from other peoples' already-recorded disks, and having a DJ "toast" over it, and recording it and selling it. This was actually the birth of hip-hop.
"Toasting" just started out meaning "to make a toast" but in Jamaica it became a part of this new form of music.
It was Jamaican DJs living in New York and playing music at parties full of Jamaican immigrants that gave rise to American rap and hip-hop. African-American people (those born in the USA) heard what the Jamaicans were doing and adapted it to their own music.
The term "rap music" comes from the USA slang expression "to rap" which means "to talk or converse" and probably comes from the word "rapport". When African-Americans started "rapping" over disks on turntables, that became what we call hip-hop.