The plant in the picture is undergoing necrosis. Necrosis can sometimes occur spontaneously in vitro as a consequence of stress (sometimes even transferring your plants to fresh medium can cause stress), but it can be also due to medium composition; please check if the composition of your cultivation medium is appropriate, also check its other properties, like pH.
If this is the first time that you're trying to cultivate that plant in vitro, you should look for optimization of the medium composition. If you have done that before and had no necrosis, then just forget about this plant and look for healthy, viable explants.
I think that the surfice humidity is too higher. IUt is also a reason for necrosis. You have to use ventilation or to put higher concentration of agar.
No neeed to worry, this type of necrotic symptom usually seen in tissue culture. Sometimes This type of symptoms seen because of lowering of pH and zink. And or increased of ethylene level in culture vessels. If few plantlets showing this symptom, then no need to worry.
I agree with my colleagues that the symptom samples in the photos, has to do with a problem in the preparation of the medium. Usually an imbalance in the components of microelements can cause color changes in the tissue until the death of the explants. If vitroplants are heavily damaged as you show, will be very difficult renew them. Therefore you should check your means, including pH and starting from new donor material.
You have not mentioned the time you have kept the explants in vitro nor have you mentioned the media composition or sterilisation method used - so the answer would be speculative - sometimes too much auxin in media may lead to necrosis - it may alo be due to the formation of phenolic compounds - if this is a regular occurance - try using activated charcoal in your medium.
You might try a control by using explabts from germinated seeds and a standard procedure as for Petunias, then try to check what was the difference in your plants step by step if petunia works. http://www.insad.pl/files/journal_pdf/journal_2010_1/full7%202010_1_.pdf
I agree with everybody who wrote that this is necrosis, not browning. If this is the initiation of a culture, this might be due to the strength of the sterilizant, and you should either lower its concentration, or use another sterilizant. If this happened during some of the passages after the culture is already initiated, this might be due to a change of the pH, or difficult gas exchange in the container. I would check the pH first, then the composition of the medium. Usually the medium for initiation is different from the one used for micropropagation. Do you add something in the medium for elicitation? Some elicitation agents cause instant necrosis, and should be carefully chosen.
if its below 10-15 days then it may be due, to the ph, or due to the sterilization procedure.. if its greater than 15 days try to subculture in fresh medium,
try adding activated charcoal if in case the phenolics are produced by the plant in the media