Hashtags serve several different functions on Twitter. I’m not sure how similar Mastadon is in this respect, but here’s some of the purposes:

Areas of interest
So if you support a particular sports team, or like a particular TV show, or whatever, you can search for (and use) hashtags around that topic to find discussion and to make your thoughts more likely to be seen by those interested. These hashtags are likely to remain relevant over a long period of time, since their subject is a permanent, ongoing thing.

Trending news/issues
When something notable is happening in the world, hashtags are a good way of related information. This could be a sports event, or a breaking news story or whatever. These hashtags are usually going to be relevant only for a limited period of time, but they’re useful for in-the-moment info/discussion.

Jokes/Messages/Emphasis
This is like an ultracompressed and more focused version of the above. So if someone posts something they think, and uses a hashtag (which changes the text colour and thus draws the eye), it can be simply to highlight that message, a bit like using bold or italics. This is only likely to be relevant to the individuals involved in the conversation, although followers may get something from it too. They’re usually intended for one-off use, not for long or even medium term grouping of ideals or information - although their use can also be part of a wider trend.

See, for example, instances of people adding #OKBoomer onto a tweet - it’s adding tone of voice and attitude to the main tweet, but it also ties into a wider trend of similar messages relating to intergenerational conflict or disagreement. It can also be used in an ironic way, referring to the more earnest usage of the tag, in order to make a humorous, or self-deprecating point.

Overall hashtags can be very versatile and useful. And even fun.

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