The proposal is to prohibit fractional reserve banking. So if enacted, Icelandic banks would have to have at least as much in reserves as they have in checking deposits. To do any lending, the bank would need to raise funds either from it's equity holders or by borrowing from investors. A number of US economists, such as Chicago economist John Cochrane, have endorsed this idea.
The main reason for abolishing fractional reserve banking is to eliminate the potential for bank runs. Standard banks don't keep enough money in reserve to pay out everyone's deposits, but they are obligated to let anyone withdraw their deposits at any time. Thus, if too many people withdraw at once the bank can go bankrupt, and that possibility of loosing their deposit in turn causes everyone to try to withdraw at once. It's a self-fulfilling panic that can happen at anytime, and doesn't necessarily even need to be caused by anything in particular. In economics terms, fractional reserve banking has two equilibria at all times: the normal one with deposits and lending, and the bank run equilibrium.
The US and most countries have tried to elimintate the bank-run equilibrium by insuring deposits--this is what the FDIC does. The problem, as countries like Cyprus found, is that if depositors don't believe the government really has the resources to pay out the full insured deposit amount for everyone, then you can still get a bankrun, this time not merely wiping out the banks and everyone's savings, but also the government's finances as well. Small countries that can't borrow and lend in their own currency are especially vulnerable here. Iceland does have its own currency, but often does business in US dollars or Euros anyway.
Additionally, there is concern that fractional reserve banking could lead to increased risk-taking by banks, since part of the risk is born by the depositors who (it is assumed) aren't as sophisticated as equity investors at measuring the banks' risks.
The drawback, of course, is that this likely makes checking more expensive to depositors since the bank can no longer lend the money out to make interest. Presumably it won't lead to higher borrowing costs because the central bank would offset the inability to lend out deposits with a larger base money stock, but it could exacerbate inequality as depositors would no longer share the returns to banks' investments (think Mr. Potter vs Bailey Savings and Loan from the movie "It's a Wonderful Life").