How do I get to the next level?

Once you achieve enough points or will advance to the next level. You earn points by getting three of one color in a row.

You earn bonus points for more than three in one row; there are only three boxes per row, but if you had a small circle and large circle of the same color in one of those boxes then when you get three boxes in a row you get a bonus for the extra ring.

You also earn bonus points for clearing two rows/columns at a time for intersecting rows/columns that are cleared simultaneously by placing one matching ring in the overlapping box.

Additionally, when you get to the level with three rings in each box (or two rings and one "dot" is more what it looks like) then if you have all three rings of one color in the same box, all of the rings of that color in the entire board will clear.

I don't know the specific point amounts you need for each level, but maybe the above tips will help.