We are composite beings, meaning we are made of both body and spirit. We're not just bodies alone, and we're not souls that are just driving the bodies around. Both parts of us are important. It makes sense, then, that we can reach out to God with both body and spirit, in both vocal and mental prayer. 
Both vocal and mental prayer are important. Soul does not trump body, and so mental prayer is not somehow better than vocal prayer. It is different and gives us another way to connect with God. Fr. Thibodeaux calls this "listening to God", as opposed to talking at or talking to Him. (p. 25)
We can use our bodies and speak in rote vocal prayer or unrehearsed talking to God. Similarly, we can use our minds in a rehearsed or read prayer, or in a spontaneous, unrehearsed way. And similarly, starting with rote prayer by reading or reciting is a good first step in mental prayer.
The rosary provides a good starting point for mental prayer. Especially for those (like me) that have a hard time sitting still and being quiet, the acts of speaking words and touching beads provides a spiritual, connected activity for the body while the mind is actively working.
Any good resource on praying the rosary will list the meditations for each decade. For each decade of the rosary, create an image in your mind of that scene. If you're praying the first of the Joyful Mysteries, imagine the annunciation to Mary. Your imagination may pull up a painting or picture you've seen of it. It may focus on the whole picture or a specific element, like "zooming in" on Mary's face as she receives the Good News. It may stay focused on one picture or flicker through many, like a slideshow or movie. The goal is not to see the event the way I see it, like a spiritual paint-by-numbers, but to connect with the event in your own mind, in your own way.
On one level, you're connecting with God in a new and different way, being mentally "present" to a part of His Divine Revelation to us. On another, you're training your mind for mental prayer. While there are certainly spiritual benefits (this is, in fact, one of my favorite ways to pray), the practice of focusing your mind on the image is good mental exercise and helps ready you for deeper exploration into mental prayer.
Thibodeaux, Mark E. Armchair Mystic. Cincinatti: St. Anthony Messenger Press, 2001.