Just adding a little bit more info taking from the answer above.
You got two types of anaerobes:
Facultative anaerobe that makes ATP by aerobic respiration if oxygen is present. It can switch to fermentation or anaerobic respiration if oxygen is absent. Most common example of a food pathogen in this category: E. coli
Obligate anaerobe which are killed by normal atmospheric concentrations of oxygen. Example: C. botulinum, which is mentioned in the answer above.
I am using C. botulinum as the example here because it is widely used in making HACCP (Hazard Analysis Critical Control Point) plans. These are plans with certain steps which make sure that what you're producing is proven safe for consumption.
Now, C. botulinum vegetative cells die in the presence of oxygen. The spores allow the organism to survive when there's lack of nutrients and/or oxygen levels are too high. The spores are also heat-resistant. C. botulinum cannot grow below a pH of 4.6, so acidic foods, such as most fruits, tomatoes, and pickles, can be safely processed in a water bath canner.
If the pH is above 4.6 there's usually a CCP (Critical Control Point) in the HACCP plan to make sure the product is pasteurized. For C. botulinum that is a kill step at 85°C for 5min minimum to destroy the toxin.
Another product example would be cold brew coffee. You need a certain % of dissolved O2 so that C. botulinum doesn't proliferate since cold brew coffee has a high pH (>4.6 usually).
Another problem with Reduced Oxygen Packaging (ROP), not related to anaerobic bacteria, is mold growth. To mitigate this some products might have the air present in the bag replaced with nitrogen then flushed. Nitrogen Flushes in ROP can reduce the rate at which mold can grow.
TL;DR: They do grow but there's certain steps each company needs to take to prove safety of their product before sale. If your HACCP plan accounts for C. botulinum you're sure to knock out a wide majority of obligate anaerobes (if not all) that could grow in your product.