The phrase often describes a situation where two parties confront each other, testing who will give in first. The two companies were playing chicken, waiting for the other to withdraw the competitive bid first.
Pro Tip 2/3
Not Literal
Although it sounds physical, 'play chicken' is usually a metaphor for any risky standoff, not just in vehicles. The two politicians were playing chicken, seeing who would back down from the debate challenge first.
Pro Tip 3/3
Negative Implication
'Play chicken' often implies recklessness and is used negatively to criticize the risky behavior. He's just playing chicken with his health now, ignoring all the doctor's advice.