How to Achieve This Timelapse Effect I can't really explain it so I'm finding it hard to Google for tutorials. I was wondering how to get this effect:

Instagram Video

By using something like Photoshop CS6 or a freeware program. I'm familiar with using the timeline to create normal timelapses, just not ones where the previous images linger on. Am I right in guessing a lighten blending mode is used?
That is something called Starlapse. In short it is a form of time-lapse, as you use the same techniques to shoot it but you use a different method to produce the final video. there are many ways in doing this using different programs and plugins. All you need to do is do a search (in a very popular web search program that rhymes with "poodle" and there are hundreds of posts, tutorials, and videos on how to do this.

I do it myself and use https://www.markus-enzweiler.de/software/software.html simple and easy to use but you are limited to editing the images outside of the app. There are other options of using plugins in LightRoom, PhotoShop etc

Hope that answers your question