Resistance mutations show up on their own during DNA replication, at a low but nonzero rate, and with no antibiotic involved. A point mutation in a gene that encodes a drug target can change the target's shape enough that the drug no longer binds, while the target still works well enough to keep the cell alive. Because these mutations occur randomly, a large bacterial population will almost always contain a few cells that happen to carry a resistance-conferring change.
Add an antibiotic and it kills the susceptible majority. The survivors were not induced to become resistant; they were already resistant. They now reproduce without competition, and the population shifts toward resistance. The key relationship is that the antibiotic is a selective filter, not a mutagen. The probability of a specific resistance mutation is roughly the per-base mutation rate multiplied by the number of cells, so larger populations and longer exposure increase the chance that a resistant variant exists and is amplified.
A resistance mutation often carries a fitness cost, because the altered target may work less efficiently than the original. Without the drug, resistant mutants can be outcompeted by susceptible cells. That cost is why resistance sometimes declines when antibiotic pressure is removed, and why compensatory mutations that restore fitness matter in clinical settings.