This is part of the game, called "influence".  Your culture can eventually cause a planet to rebel and switch sides.

INFLUENCE is therefore countered with INFLUENCE.  As well, planet RESISTANCE will help your culture resist theirs.  Its very crucial in this game.  Colored hexes around your planets are Zones of Influence.  Each planet produces influence, and going unchecked by an opponent (or lack thereof) will cause ZOI borders to change.  When a planet is fully surrounded by another's influence, it begins to rebel, which again can be slowed by planetary resistance modifiers.

The game is more than just science, production, and money - it lets you play and even win with the influence game.  If you are playing passive and defensively, you certainly need to worry about influence, as this is the "peaceful killer". 

If you are not willing to go to war with an influential neighbor, you can attempt to raise the local influence modifier of your border planets so that their influence growth per turn are greater than that of your neighbor's planets. If it is not possible to build any more influence buildings on those planets, you can build and develop a culture space station nearby instead. Altarian Resistance and Iconian Refuge generally have high global influence. 

You can also attempt to up your own global influence by researching the political tree, developing large and happy colonies, as well as claiming culture relics. Relics are incredibly powerful and their bonuses are stackable. One culture relic can increase your global influence output by 30%. Too powerful some would say, as the AI largely ignores them... Large and happy colonies are great for everything.

To add to this thread. 

• Make sure your planet is 'happy'. While happy your planet has fair to good resistance. 
• Build influence buildings on your planet, these will grow influence on that planet. 
• Build Culture starbases either near the unhappy planet or on your borders
• Follow along the Benevolent Ideology which provides extra perks on Influence.

Gilmoy (one of our Elilte Alpha's here) did a thread regarding the rate and spread of influence and how it works. Influence is rather important and is difficult to beat if you never paid attention to it. 

Its growth is linear and grows more and more over time as population grows. You are always building influence but with the help of buildings it grows faster. If you colonize near a Civilization that is known for spreading influence (Altarians are Prime on this) than you better build influence buildings right at start. 

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