I've received quite a number of badges, but although I've searched in meta and in the main English site, I couldn't find what the point of badges is.
Badges reward good behavior
The blog post that first announced badges describes their purpose thusly:
Our goal with the badges is to encourage people to a) have fun and b) use the Stack Overflow website in ways that make sense.
A second blog post about badges says something along the same lines:
Badges exist to reward and encourage the kind of positive behavior we want in our community.
Badges help indicate participation
Although it's not mentioned in the aforementioned blog posts (due to changes that happened in 2015 after those blog posts were written), there is another place badge count... counts. The "Candidate Score", which is shown for each person running during each moderator election, is based on a combination of reputation and certain badges:
- 1 point for each 1000 reputation up to 20,000 reputation for a maximum of 20 points.
- 1 point each for Moderation badges - Civic Duty, Cleanup, Deputy, Electorate, Marshal, Reviewer, Sportsmanship, Steward - for a maximum of 8 points.
- 1 point each for Editing badges - Copy Editor, Explainer, Organizer, Refiner, Strunk and White, Tag Editor - for a maximum of 6 points.
- 1 point each for Participation badges - Constituent, Convention, Enthusiast, Investor, Quorum, Yearling - for a maximum of 6 points.
For badges that can be awarded multiple times only 1 point is granted for each badge type, thus ensuring a maximum score of 40 points.
People of course are free to vote however they want, but higher scores look better.
On sites where users leave "welcome" comments for new users, it is often suggested (for example here) to only suggest a new user take the tour if they don't have the "informed" badge (which is given for taking the tour).
Badges also give a quick indication how much a user has participated on meta, which doesn't have its own (visible) reputation system.
They are akin to getting "achievements" in computer/console games. Badges are one of many ways that Stack Exchange games you to participate more in it, as are points, privileges and the like.