"Crashing" in this place means "staying/sleeping at".
"The Y" is shorthand for "the YMCA" - famous from the song... the "Young Men's Christian Association".
From the Wikipedia article:
From its inception, it grew rapidly and ultimately became a worldwide movement founded on the principles of Muscular Christianity. Local YMCAs engage in a wide variety of charitable activities, including providing athletic facilities, holding classes for a wide variety of skills, promoting Christianity, and humanitarian work. YMCAs continue to be religious organizations; many national or local organizations de-emphasize this aspect, while others choose to prioritize its religious mission above all others.
The humanitarian work has included offering a place for someone in need to stay for the night or get a hot shower or meal.
Their current logo and website actually minimize the full name of the group, and depict the current term used for the association "The Y".