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I am new to sous vide food. I am cooking a chicken breast. I have read that chicken requires an internal temperature of 165 Fahrenheit. However the recipe only requires a temperature of 150F.

https://recipes.anovaculinary.com/recipe/chicken-breast

How does this not cause food poisoning?

We checked the internal temperature after an hour and it was only 130F. Is this safe to eat?

The reduction of bacterial growth, and thus food safety, follow a logarithmic pattern that factors in temperature plus time. During sous vide cooking, lower temperature are frequently used for longer times. Employed correctly, this renders food safe. For an excellent explanation see the work of Douglas Baldwin.

The link says for "soft" chicken you should let it go to 140F for 1.5 to 4 hours.

You checked at 1 hour and it was below that (130F) ... so you need to cook it some more.

This link gives the same temperature for chicken and explain why the lower temperature is acceptable when cooking for longer time

Edit: Good link for the temperature safety thing.

The other answers are correct regarding why that temperature is okay for sous vide, but I just want to clarify why it's not good enough when using other cooking methods.

As moscafj suggested, you need the meat to spend a certain amount of time at any particular temperature to actually kill off enough pathogens, and this is where sous vide acts quite differently from most other cooking methods.

The key difference is that sous vide never exposes any part of the meat to higher temperatures than the target temperature, so you get a lot of time right near that temperature; whereas more traditional methods typically involve exposing the meat to far higher temperatures, so the meat will be shooting right past that minimum safe temperature in a short amount of time. For example, when cooking on the grill, it might be exposed to air that's 450°F or more, so if you measure the meat at 165°F, it might have only been at a safe temperature for a minute or two — which is just enough time to kill those pathogens. (Also note the carryover effect.) If you had pulled it off the grill at 150°F, it would only have just entered the safe region, so few of the pathogens would be killed before you start cooling it again. On the other hand, you can't leave it on the grill for much longer, because it will quickly attain the texture of cardboard. Sous vide will never pass the target temperature, though the texture will degrade if you wait a really long time.

All that being said, sous vide does take a bit longer to get the meat up to the target temperature in the first place — and the colder or thicker your piece of meat when you put it in, the longer it takes. Even if your water temperature reached 150°F a minute after you put the chicken in, the meat itself will be far below that temperature for quite a while. So no, chicken that's at 130°F is still coming to temperature, and you have to cook it for longer because it needs to actually spend time at your target temperature. It may only need around 6 minutes at 150°F, but it needs to get there first.