There are two ways:

1. Station capital ships at the affected planet. Each capital ship repels culture depending on its level (and researched bonuses in the case of the Vasari caps). Culture repulsion rate can be calculated by 0.05x + 0.25, where x is your capital ship's level, if the capital is at friendly location. If you are at a neutral location, divide the result of the calculation by 2, and if your capital is at a hostile location, culture repel rate is zero.

Vasari Capitals can have their repulsion rates increased with Shock and Awe research, providing a 13% bonus to repulsion when Shock and Awe is at level 1, 25% when Shock and Awe is at level 2. The repulsion bonus is added after repulsion rate is calculated from the expression above.

2. The other method is to build culture centres of your own to resist the advance of hostile culture. Effects of multiple culture centres stack. You can tell if your culture is stronger or weaker than a hostile culture by looking at the coloured phase lanes - if yours is advancing, your culture is stronger and vice versa.

Sometimes you have to go hunting for the enemy's culture centers.  If I have to build more than a couple to ward off culture, I start looking at military options and bringing my main battlefleet into play.

If you are in military deadlock, you have to build more culture stations at the threatened world AND neighboring planets behind the one being threatened to help push against the enemy culture.  In this situation, researching the culture techs also makes a big difference because it is amplifying the power of all the culture stations to win the culture war. 

This doesn't happen much in my experience, because normally someone has military superiory.  Why do I want to build a bunch of pretty culture stations when I can build warships?  But in those few situations where your military is tied up elsewhere or you are on the defensive, tech'd up culture stations will save your ass.

Early in the game, you should try to complete the AI mission and attack one of your AI opponents. Normally, you will earn a cease fire by this way. Then continue to build up your force and attack you enemy until you are victorious.

Optimum number = number of colonizable planets you can sustain

This number is not constant, as it fluctuates according to your resource demands, fleet size (and capacity), proximity to hostile strongholds, available resources, and numerous other factors.

You should only colonize a planet once you can afford to upgrade it to generate positive income, otherwise it becomes a drain on your resources. Defenses should scale according to the threat you percieve is posed by Pirates or opposing factions (all you need is to buy enough time for a fleet to save the planet). Remember that increasing fleet capacity imposes a percentage reduction on your overall income, so upgrading too quickly will cause your economy to collapse and make it significantly more difficult to raise income rates.

In short, keep expanding, but at a pace which doesn't cause your economy to suffer or spread your fleets out too thinly.

I was getting demolished by a hard AI's culture spread and I couldn't penetrate as his choke point planet was well defended.  This one planet was the only way in or out of his empire.  Two of my planets were flipping so I took about 20 scouts andI distracted his garrison fleet by launching a few units to one side of his entry phase lane while i took my scout fleet made a timed explosive suicide run through the contested planet while my diversion was in play and i got my scouts through his bulwark desfense and they eventually took out over 10 broadcast centers on 6 planets.  It was epic.  My existing culture began pushing out and soon i had this enemy empire on the ropes thanks to a great decoy.