The expression means "you can rely on that". It comes from writing paper checks. If a check is written and the payer does not have enough money to cover the amount written on the check, the person trying to cash the check will have to pay a fee due to the check "bouncing" (this is where the term "rubber check" comes from). If somebody tells you something and then says "and you can take that to the bank", it means they are confident in the validity of their statement. A similar phrase would be "You can bet on it." This phrase can be used in day to day speech today, but I'm not sure if the phrase will survive another generation due to the use of checks in much decline. Hope this helps :)

"You can take that to the bank" means that whatever was being said or promised is certain. For example, Joe might say "If you get an engineering degree, you'll have no trouble getting a job. And you can take that to the bank." This means that Joe thinks it's guaranteed that you'll be able to get a job. This is an older expression which I don't hear used much these days. I hope this helps!