A "poster" is a picture, usually an advertisement. A poster is printed cheaply. Hundreds of copies are made. They are intended to be "posted" on walls, perhaps outdoors.
You are doing something "spontaneous" when you do it without a plan. You do it out of impulse. You suddenly feel like doing it, with no reason, so you do it.
I walk down the street. It is a nice day. I feel good. Suddenly I realize I am whistling a tune. I didn't plan to do it. I just started whistling spontaneously.
"Bach's music sounds carefully crafted. Gershwin's sounds as if it just poured out, as freely and spontaneously as the song of a nightingale."
The writer is saying that Weingart designed posters that looked spontaneous. They didn't look as if he thought about them. He just said "Right now, I feel like putting this shape here, and that color there." They looked playful. They looked like the artist was having fun. They looked "chaotic," as if he was just putting things down on paper, without a plan. They looked like the spontaneous product of his inspiration.