I found numerous videos on Youtube where non-Muslims embrace Islam and become emotional.
What is that emotion about? Why they cry?
Since i'm from those who accepted Islam (Alhamdulillah) i can answer from my part.
As a Christian back then i was in and out of Christianity due to doubt and other things which led me to sins. I never felt that Jesus (a.s) could of been a God or part of a trinity concept. So i was sort of rebellious already in my belief. But i always thrived for a pure monotheistic religion which (at the time) was very hard to find. I searched from the Jewish and some denominations (sects) within Christianity for a pure way of life i had even started to practice prayer again and even slowly practicing Kosher laws which is similar to Islamic dietary laws. But within me i always felt a hole in my heart like something essential was missing and not Judaism or Christianity could fill that hole.
They also advertise the fact that since i did certain sins i could never be forgiven.
To which when i found out about Islam and read the very first Surah, I cried instantly.
Surah Al-Fatihah, emphasise the mercy of Allah (swt) which i strived to search for since i became religiously aware. And when i accepted Islam this hole in my heart was filled.
So yes to best illustrate it is the Qur'an itself.
وَإِذَا سَمِعُوا مَا أُنزِلَ إِلَى الرَّسُولِ تَرَىٰ أَعْيُنَهُمْ تَفِيضُ مِنَ الدَّمْعِ مِمَّا عَرَفُوا مِنَ الْحَقِّ ۖ يَقُولُونَ رَبَّنَا آمَنَّا فَاكْتُبْنَا مَعَ الشَّاهِدِينَ
And when they hear what has been revealed to the Messenger, you see their eyes overflowing with tears because of what they have recognized of the truth. They say, "Our Lord, we have believed, so register us among the witnesses"
is one such ayat.
And when they (who call themselves Christians) listen to what has been sent down to the Messenger (Muhammad SAW), you see their eyes overflowing with tears because of the truth they have recognised. They say: "Our Lord! We believe; so write us down among the witnesses. (Qur'an 5:83)
This verse is for the Christians. You can read more from here.
But, I guess, most of the people whom you've seen crying were Christians once, and/or are the people who've truly recognized the truth and are guilty of their past.
May the creator guide us all.