How could antibacterial resistance rapidly spread in a bacterial population being treated with an antibiotic?

It spreads by conjugative plasmid.

Explanation:

Bacterial population spreads their antibiotic resistance by conjugative plasmids, a circular DNA with locus for partitioning and origin of replication. Conjugative plasmids exist naturally in bacteria. Normally they multiplyand live inside the cells, using their synthetic machinery, but can also transfer its genes to neighbouring cells. It is this characteristic that causes rapid antibiotic resistance in bacterial population even if those bacterial species are not related to each other.