Hashtags are words that start with '#'. In the case of Groups.io, they only apply to the subject of a message. So, if you send a message to your group with the subject 'This is a sample message #test', the message will be categorized with the hashtag '#test'.
You can search the archives by hashtags. And you can set hashtags so that any messages tagged with them will be deleted automatically after a specified period of time.
Hashtags are completely optional.
One potential use of hashtags are for intros. Tell people when they join your group that they should post an intro message about themselves, and have them tag it with #intro. Then, you can search on the #intro hashtag and see everyone's intros.
Another potential use is for neighborhood groups with people offering things for sale. Those messages can be tagged with #forsale, and that tag could be configured such that messages tagged with it are automatically deleted from the archives after a month. So you're not left with a bunch of for sale messages in the archives that no longer apply.
An advantage of tagging messages is that members can mute hashtags that they're not interested in. If someone is not interested in the #forsale tag, they don't ever have to see any of those messages.
Your group can be configured to require all messages to be tagged or not. You can configure a hashtag so that only moderators can post using it. Or you can configure a hashtag so that messages tagged with it will not be sent to the group. This is useful for 'loading' messages into your group. And you can configure a tag so that messages tagged with it will automatically be deleted after a specific time.