To claim the earned income credit, you must first claim the children as dependents. This means the children must live in your home more than half the nights of the year. If the children also lived with their parents more than half the nights of the year (because you all live together at least some of the time) then the parents get first claim and must agree to allow the grandparents to claim the children.
Then second, to get the earned income credit you must have earned income. Earned income is income earned from working (performing a service for money.) Unearned income, which include things like retirement pay, most rental income, dividends and capital gains from sale of stock, gambling prizes, and so on, doesn't count for the EIC. And in fact, if your unearned income is too high, then you are disqualified from EIC even if you also have earned income.
The concept is that EIC is intended to help young (poor) working families who are starting at the bottom of the income ladder. If you have large amounts of unearned income from investments or rental property, then you probably aren't "poor", and if you aren't "Working" to get earned income, then you don't get a credit designed for workers.
(To go into even more detail, EIC started out as a rebate of social security tax, because even though the income tax is "progressive" and the poorest hard working people might owe no income tax, social security tax is "regressive" because every pays the same, even the very poor, and high income workers pay less. So EIC was a rebate of social security tax based on the concept of earned income. Unearned income is not subject to social security tax, so it isn't eligible for the rebate.)
