"the depth of the Depression" - during the lowest point of the Great Depression. There were very few jobs, and money was scarce.
"by the hundreds" - this is just another way of saying "hundreds of actors were out of work"
"Pop heard me out" - His father stayed quiet and listened carefully to everything that his son had to say.
(Note: Usually when we say "hear me out" to someone, it's because we are telling him something that sounds impractical or unreasonable. For example: "Dad, I want to quit my job. Before you say anything, hear me out. I've got a lot of money saved, and my car is paid for. I'd like to take a year off and travel across the country."
"Stretch their wings" - try new things. Go beyond your "comfort" zone - visit new places, experience different types of activities.