Catholics believe that Christ unites his people on earth and in heaven. The saints intercede for the Church through Jesus, the one mediator. Asking for their prayer is a way of sharing in that communion.
A little more context
A saint is not a rival to God or a power assigned to deliver a particular result. Patronages help us find companions and examples, but no prayer guarantees a cure, job or other outcome. Devotion bears good fruit when it draws us toward Christ, the sacraments and love of our neighbors.
A practical beginning
Choose one saint whose life interests you. Learn what that person actually did, then identify a virtue you can practice. Ask for prayer in a short sentence rather than worrying about finding a perfect formula.
Source & further reading
Catechism 946–962: the communion of saints (opens in a new tab)↗Teaching summaries are introductions. Follow the linked Church source for its fuller context.