The idiom 'stick your neck out' is used when you want to talk about risk. It is very popular, and you will hear it used (and you can use it) in both social and professional contexts.
As you can see from our video, it is mainly focused on the person speaking, e.g. "I stuck my neck out for...", "I'm tired of sticking my neck out for...", etc.
However, you can talk about someone else and say something like, "He stuck his neck out to defend your proposal". The purpose remains the same. In each case, the person 'sticking their neck out' is taking a risk that their action(s) could lead to trouble for them, or some other generally negative outcome. Remember that the subject of the idiom (I stuck my neck out, He stuck his neck out, She stuck her neck out, etc.) is the person taking the risk.
Most of the time, using this expression implies that the person taking the risk doesn't necessarily trust that things will turn out the way they expected - but that is exactly the point of taking a risk! If they imagine that things could go bad, this might make them angry, so this idiom is often used angrily: "I can't believe I stuck my neck out for you! Now I'm in trouble!"
You could also use this idiom to provide advice or to motivate someone (although this is not a very common usage). For example, you might say to someone, "You should try sticking your neck out a bit more often. Speak up for yourself or put yourself forward for promotion".
Phrases with a similar meaning include 'put your head on the block,' 'put your neck on the line,' and 'go out on a limb.'