Some speeches are purely for informational purposes. You tell the boss and other executives about your portfolio of projects. You brief students on what to expect during their summer abroad.
Other speeches, including speeches that are heavy on numbers, could be livened up with storytelling. Don't assume that a story has to be a long, fictional tale that illustrates a theme or communicates a moral. That's fiction writing you have in mind.
An anecdote, a short story that draws on real life, can liven up a presentation and make a point. You can weave anecdotes into any sort of speech. In some scenarios, like a project briefing, it can be tough to find a place for an anecdote or two. Use the topic and the audience to decide whether in makes sense to try. Some listeners might like to hear how you identified and solved a problem. Most managers and executives will just want the facts.
In most speeches anecdotes are integral to the material. Are you telling people about your year abroad as a visiting scholar? They'll want to hear stories relating the most interesting or important things that happened. This is not the place for a dry summary of what happened and when.