Team Fortress 2: What is the purpose of "Support Points"?

The "purpose" of support points is the same as the "purpose" of other stats like damage: support points provide an indicator of how much you're supporting your team. They don't contribute to points directly, however most support actions also provide bonus points.

Based on the scoreboard page of the wiki, these are the most common ways to earn support points (this is not an exhaustive list, see the wiki page for details):

• Kill a medic with full uber: 500 points
• Heal allies with your mad milk: 1 point per damage done by allies
• Extinguishing a teammate: 250 points
• Minicrit assists (Jarate/Fan/Banner): 1 point per 4 damage done by allies
• Destroy a sapper: 250 points
• Giving a sandvich to a teammate: 250 points

Since in your screenshot you're a pyro, I anticipate many of your points came from either extinguishing teammates or killing medics with full uber. If an enemy medic is using the Vaccinator, you could quickly rack up support points because a medic at 25% Vaccinator charge counts as full uber.