TL;DR: Time series data are measurements ordered by time. Time series analysis tools like Prophet tries to guess future values based on past ones. This matters because many economic decisions take future values as inputs, and we can't measure the future values yet.

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Time series are data values ordered by time. So amount of rain for each year 1993--2017, or the sales each week of Q1 this year, or service time for each request received the previous minute, or ... you can come up with your own examples.

The reason for ordering by time is that we think that there might be patterns in the data that are revealed over time. The measurements might be evolving in (somewhat) predictable ways, day by day, or year by year.

Say, then, you know how much sunshine your area received per year between 1999 and 2020. You're thinking of installing solar panels. Then you have an interesting problem at your hands: given what you know about sunshine historically, how much sunshine will you receive in this year, 2021? How about 2022? What can you reasonably expect to get out of your solar panels in the future?

This is probably the most obvious way to apply Prophet: I have an economic decision to make, which takes as input future data. Obviously I don't have future data yet, so I need a good guess based on the historic data I have. Prophet attempts to make that guess.

Time series are numbers measured or aggregated at some consecutive times or days. It's a data type with a specific structure, like a set, a vector or a matrix: what's inside varies a lot and depends on the specific application (e.g. daily temperature, rainfall, stock prices, items sold, water level in a river, corona infections).

Prophet is a Python library that implements a specific type of extrapolation model to predict the future: i.e. you input a few months/years of historical daily sales figures, it tries to detect trends and recurring patters, and it outputs predictions of future daily sales. It's currently one of the most popular approaches, probably because it's pretty easy to use and a good fit for many time series in commercial companies. The article shows a lot of of examples of time series where the approach does not work well.

If the layperson doesn't have to produce forecasts for their job, (s)he probably has little use for Prophet (or other statistical forecasting models).

If you have historical data with timestamps and you want to predict how that data trends in the future, you can apply statistical methods yourself, or train a neutral net, or use an off-the-shelf package like Prophet.

Whether a layman finds it useful depends on the data, what they're used for, how well they're suited to Prophet's strengths, how lay the man is. I imagine by the time a layman gets to the point of understanding its strengths and weaknesses, she will no longer be quite a layman