Listen to ska and reggae music. If possible, count along with the downbeats and offbeats to achieve the proper feel of the music. Typically in music, you can count "1-2-3-4." To count the offbeats, you add "and" in between each beat, making it "1 AND 2 AND 3 AND 4 AND." The "and" offbeats are the beats that receive emphasis in reggae music.
Listen to the trumpet's role in the music. Horns in reggae are used typically for introductions, counter melodies and solos. They are used softly throughout the song while another instrument is soloing or if the vocalist is singing.
Learn common reggae chord progressions in a variety of keys. The tones of these chords will most likely be the notes the trumpet will play. Most chord progressions will be I-IV-V-I or a slight variation.
Play along with a recording of a favorite reggae song. The trumpet will mimic the offbeats played on the guitar while the vocalist sings but may break away for a solo. The solo is likely improvised, so practice with improvisation can be helpful. To practice improvisation, first learn the scales that are used in a particular key. For instance, learn the major scale in the key of Eb major. Use the tones in the scale for your improvisation, coming to rest on the important notes of the scale, those being the tonic (Eb), third (G) and dominant (Bb). See Resources for a site with MIDI tracks over which you can practice soloing.
Look at some trumpet tabs on the Internet for additional help with soloing and learning riffs on the trumpet from popular reggae tunes.