I grew up in a family hot air ballooning business, and while I haven't been involved in a few years, I can answer your question in two words: They don't!
As far as collisions go, the other answers have mostly addressed this: hot air balloons only control vertical movement, so with all the balloons experiencing the same wind, they'll move at approximately the same speed.
When taking off, we always angled the balloon so that the wind would blow from basket to envelope. In a large group, such as at festivals, we would maintain the same practice as much as possible, and have a line of balloons perpendicular to the direction of the wind. Often, this would restrict the space enough that inflating balloons would roll slightly into adjacent ones, but with a competent/strong person on the crown-line (the rope attached to the top of the balloon, used to maintain stability in the event of crosswind), the small gusts which caused localized rolling could be overcome.
In large groups, when a balloon is ready to lift off, the pilot will typically ensure nobody is going to be directly overhead, then apply more heat to lift out quickly. This prevents the basket from catching any of the envelopes of the balloons still on the ground. As mentioned, once in the air, collisions are not really a problem.
EDIT: I believe you can see two balloons on the ground but vertical which are pressed up against each other in the picture you've provided. Try looking near the middle, just to the top left. You can see that the envelopes are harmlessly pushing away from each other while the baskets are still quite far apart.
In The Air
Balloons are huge and move slowly, particularly slowly relative to each other. If two balloons are exactly level and one directly blocks the wind to the other when a gust comes, it's possible for one to suddenly gain speed. I've never seen gusts strong enough or long-lasting enough to create a speed difference of more than a couple miles per hour, so when the balloons do touch, the envelopes gently 'kiss,' then bounce back. This can, at worst, create a slow rocking of the basket.
Now, the last situation, if a pilot is unaware of another balloon directly overhead and starts going up, it could be a problem. However, pilots typically have radios and can communicate with each other if it's necessary. The pilot in the unseen balloon above would start applying heat to try to get out of the way as well as radio some (likely unkind) choice words to the pilot below him.
This is more of an after-thought, but if a pilot wants to land and another balloon is already there, the pilot will simply stay in the air a little longer to find another landing site.
A second after-thought, but I know about 10+ years back there were a couple competitions at festivals where pilots would receive a special ribbon device and a partner-balloon. The device would mark the longest the ribbon had to extend. Each of the pilots would attach one end to their basket, then the duo who stayed the closest to each other would win a prize.
Another type of competition also occurred in which the pairs of pilots would take off about 15-30 minutes apart and the second pilot would have to try to land as close to the landing site of the first one as possible. Sometimes, this meant landing at roughly the same time and in uncomfortable proximity. Safety is always the first concern of a good pilot, though! If a pilot wasn't sure it could be done safely, he or she wouldn't do it!
As longwinded as this is, the gist of the answer is that balloon-balloon collisions aren't very serious unless the basket is involved. Typically, they move slowly enough to make it easy to avoid unwanted collisions, despite their lack of horizontal control.