I found a few example phrases from Danish media to help clarify and explain.
"Han fik mig til at græde" -> "He made me cry"
"Google tvinger folk til at se reklamer" -> "Google forces people to watch ads"
"Vi kommer ikke til at se hinanden igen" -> "We will not see each other again"
"til at" doesn't really belong together. "at" is part of the verb that's following. "at græde (to cry)", "at se [...] (to watch [...])" and "at se (to see [...])".
"Han fik mig til ..." -> "He made me ..." what did he make you do? Cry, but you need to use the basic form in this case, "at græde" - "to cry".
You will use this form in cases where something cases an action on something else.
Hope this made some sense :)