The French verb rester (to stay) is a regular -er verb. It is conjugated in the usual way:
je reste---I stay
tu restes---you (familiar, singular) stay
il/elle reste---he/she/it stays
nous restons---we stay
vous restez---you (formal, plural) stay
ils/elles restent---they stay
The French proposition avec means "with". So "rester avec quelqu'un" means in English "to stay with someone". That is the denotation of the term.
The Kreyol language of Haiti has an enormous amount of French influence, but it is not strictly derived only from French--many African languages as well as English have influenced the vocabulary and the grammar. Still, there is a great deal of overlap between Kreyol and French, and Haitians still study French in school as well. So it is not surprising to find many of the same words in French and in Kreyol.
"Connotation" is different from "denotation", in much the same way that "term" is different from "concept". The denotation of a term is its dictionary meaning: the denotation for "dog", for example, is "a member of the species Canis familiaris". "Connotation" includes the emotional and other psychological layers invested in a term, thus going beyond the simple denotation. A person who has been attacked by a dog may think of dogs not only in their dictionary meaning, but as "a vicious animal"--for him or her, that is the connotation that "dog" holds.
The French "reste avec" (spelled "restavèk" in Kreyol) thus denotes that [someone] stays with [someone]. However, the reality (connotation) of "restavèk" is child slavery. A child is sent away from his or her family to "stay with" a host family.
These children act as live-in domestic servants and rarely are sent to school, receive new clothing, are allowed to play or visit their families; are often verbally, physically or sexually abused, and are nearly always exploited beyond the limits of national and international standards for the rights of children.--http://www.nchr.org/hrp/restavek/overview.htm
Often, they die young.
Some sites for learning more about the problem of restavèks and what is being done to help them are here, here, here, here, here, and here.
Je reste avec ma famille.
Tu restes avec les tiens.
L'enfant reste avec qui?