A 'lifeline' is a rope used for saving people in danger, especially at sea. It is also something which someone depends on completely.
The people who talk on this clip claim that the unemployment benefit is their lifeline, and demand its duration to be extended by the US government.
Beyond the dramatic personal and social situation of the video, so much so for its universal value and relevance in today's crisis-hit world, the video may help us get familiar with some vocabulary to do with unemployment:
to be laid off (=to lose one's job); to be out of work; unemployed worker, unemployment benefit, unemployment rate; the benefit will run out (expire)
It is also good to practise figures and get to know the names of jobs and people's everyday bills.
Anyway, if you want to understand everything, here's the script.