A "tough" person is someone who is not easily defeated, someone who is resilient or persistent. "Tough" can also mean difficult when applied to a situation (e.g. "It was tough to open the jar because the lid was stuck.") "The going" in this case refers to living through a period of time, so "When the going gets tough" means "When you are living through a difficult time". "To get going" here means to get active in some way. "The tough get going" in this case, then, means "Those who are not easily defeated get active in trying to solve the difficulty".
"When living through a difficult time, those who are not easily defeated get active in trying to solve the difficulties, rather than just giving up".