Once you have unlocked the Library, you will start seeing meteorites falling down onto the battlefield. They will appear once every minute, and just like gold they are automatically collected. Offline rewards will give you meteorites as well, up to 10 hours, so 600 meteorites. There are various ways to get more meteorites. You can use gold items and there are certain upgrades which increases the chance for 2 meteorites to drop instead of 1.
When using / spending your meteorites, your first priority should be unlocking tier 2 and tier 3 equipment on your heroes. The remaining meteorites can be spent inside the library on meteorite research. You can also unlock a second research slot inside Firestone Research, however I suggest saving up for this once you have a lot of Fire Tokens saved up, like 30 or so.