Having a narrow chest with a protruding breastbone - alluding to the appearance of pigeons.
This phrase originated in the form of pigeon-breasted, which is known since the early 19th century; for example, the 1808 edition of A Universal System of Natural History refers to "The pigeon-breasted salmon" - which must have been a strange beast.
Pigeon-chested followed later in the century. The first example I can find is from The Times, June 1873:
"He described him as thin, pigeon chested, with dark hair, perfectly straight."