To stick to something is to keep to a plan or a promise. No wandering off, whatever else happens.
It covers diets, budgets, timetables and stories, and sticking to your story is the phrase every police drama uses. There is a warm sense of loyalty in sticking to somebody. The imperative stick to what you know is standard advice, sometimes kindly meant. And a physical sense underlies all of it, since something sticky stays where it is put.
1phrasestick toTo keep to a plan, rule or promise.
“Try to stick to the budget this time.”