This is achieved when the vocalist finds the "break" in their voice. The difference between "chest" and "head" voice. Sometimes it's called a "yodel". Basically it's where the break in the vocal cords between full voice and upper octave.
Additional info in response to comment question:
The simple answer is: find where the vocalist's natural vocal break occurs, and ask them to incorporate that break into the melodic progression. Finding the break is done most easily by simply having the vocalist sing - starting in their lowest octave, and sliding up to their highest note. When you hear them switch from model voice to falsetto, you've found the break. It will sound something like a "yodel".
One problem though.
Most of our singing is done in the Modal voice. This is where the optimum balance between airflow and glottal tension occurs, resulting in maximum vibration. Modal voice is the voice we speak in.
If you want to coach someone to "break" over a note, they might be limited by their natural vocal range. If the break you want is over a pitch where the lower and higher notes are both in the singer's modal voice, they simply might not be able to do it.
In this case, try modulating the song into a higher key, so that the vocalist can incorporate their natural vocal break into the melodic progression.