It implies a decline from a previously good or respectable state to a state of disrepair, chaos, or ruin.
Here are some examples:
* "Since the new manager arrived, the company has been going to the dogs."
* "The neighborhood used to be so nice, but it's gone to the dogs in recent years."
* "My car is going to the dogs - the engine is making strange noises and the paint is peeling."
Let me know if you would like to explore other idioms!