How do you come across as spontaneous?

The best way to appear spontaneous and natural for a big audience in a public speech, is to use a speaking schedule. It is not a fully written text, but, a reflection in keywords of the content, structure and direction of your performance. A suitable speaking scheme has three columns. 

The first column,the left, is called : the timeline. There you can accurately measure how long each part will last, and how much time you should set aside for digressions, questions, or interaction with the audience. The second column,the middle one, contains the substantive keywords that support your story. If you don't feel completely sure, it helps to write out the intro and the ending verbatim. This ensures a chiseled opening and closure. The layout of your schedule is well-organized and clear. It is especially necessary that you can pick up the thread of your story at a glance. The secret here is that you have to practice the logical and verbal transition out loud. If you can do that effortlessly during a rehearsal, it won't be a problem in real life. If not, you need to intervene, change the keywords and start practicing again. The third column,the right one, is the direction column. It states what to do, where to stand, how to hold your body. It often suffices to list things in this column for you to apply them to, or omit them from, your performance. 

"All good speakers started as bad speakers." - Ralph Waldo Emerson