Jay is given antibiotics for an infection. soon he feels better, so he doesn't finish the full course of antibiotics. How may this lead to the development of antibiotic-resistant strains of bacteria?
Some bacteria are left in his body,which will find ways to get resistant against the antibiotic.
Jay may still have some of the bacteria left in his body. Even though he is feeling better it does not mean that the bacteria which caused him to feel sick in the first place is gone. Bacteria if any left within the body can find ways to get around the antibiotic, so the remaining bacteria will try to evolve mechanism to get resistant against the antibiotic ,and will trigger a new infection and this time the same antibiotic will not work as the bacteria have found a way to become resistant to the antibiotic.
Again if any person receives infection from Jay with these bacteria,he/she will also have these resistant strains within his/her body,so the antibiotic will not act for his/her case well.Thus it may spread from one person to another and can create epidemics.
I hope this helps :)