The aim of the game is to keep your plant as happy as it can be so the short answer is you can earn points by caring for your plant. Determining how your plant is feeling is done by the Solis sensor, which gives you direct feedback on how your plants is doing in real life and by extension also how your plant is doing in the game. The more consistently you care, and thus the more days in a row your plant is happy, the more points you can score and the higher on the ranking your plant will be.
Things like the number of days your plant has been happy, how many times in a row you watered your plant without missing a day, adding extra information about your plant or engaging in our online community will earn you badges that you can show your friends, but that also help you rack up points faster. How your plant is doing also determines which (virtual) bugs you attract and how fast. Bugs are an important part of an ecosystem and can actually make a plant healthier, attracting certain kind of bugs will therefore help your plant earn points.
faster.
Specific tasks will also earn you Solis coins, which you can trade in for in-game boosters that f.e. double your earned points or attract certain bugs quicker. However, if your plant is being neglected, you will lose points and your place on the ranking no matter how many boosters you buy!