Simply put, "They get the benefit, the rest of us pay the price." A "tab" is in fact a tally that the barman runs at the bar for food and drinks you have had. To pick up the tab originally means to pay the bar bill or the restaurant bill. By extension, it means "pay for something". If you drink at a bar, you can pay per drink, or you can alternatively ask the barman to "run a tab" or "run a tally" so that you only have to pay once at the end of the evening, instead of having to pay five times for the five drinks you have had. If a friend comes in, you can buy him a drink by telling the barman, "Put his first drink on my tab."

It means to pay for something