“proceeds from them” = the amount in all of your accounts and the accounts of your relatives (no definition given here but presumably they mean your dependents).
“total income … for three years” = you and your relatives/dependents income for the last three years.
“X exceeds Y by more than ₽10,000” = X > Y + ₽10,000.
Let’s make up some numbers and see where they lead. Suppose that you are a middle–aged civil servant and your salary has been ₽100,000 for the last three years. Y = ₽300,000, obviously. You’ve been saving for your whole life, so the sum total of all of your accounts, X, is ₽500,000. Most of that is in whatever the Russian equivalent of pensions or 401ks are, so they have been earning money for you. Either collecting interest, or by gaining value on the stock market or whatever. Let’s say that your average growth over the last 3 years was 5%. That means you earned ₽68,000 in interest over those three years. Assuming that this adds into Y, your total income, then Y is ₽368,000. Well, ₽500,000 is certainly larger than ₽378,000, so your name comes to the attention of the prosecutor.
However, I disagree that this means that you automatically lose anything. The article says: “In order not to lose money, the official will have to prove the legality of their appearance on the account.” Presumably then you won’t lose anything if your records show that you obtained it all legally.
On the one hand, you do want to catch corrupt officials who have been taking bribes or kickbacks. On the other hand, choosing this means to do it seems counter–productive, since you have to expect that everyone who is saving up for retirement in some way will qualify for investigation. Since the criteria for investigation are not correlated with the crime that they are trying to detect, they are going to have all the false positives.