Is the at fault drivers insurance required to pay me for the lost wages I lost due to the collision.

You may be entitled to lost wages (assuming you can prove them sufficiently--i.e. have enough wage history to demonstrate reliably what you would have earned) as a foreseeable (reasonably likely) outcome of the accident for a "reasonable" period of time--the length of time it would reasonably take someone to get their car fixed and/or rent a short-term placement. (If it takes you longer than normal or reasonable, you don't get your full amount; they are not responsible if you have unusual delays or problems). It's not a given or guaranty you'd get this if you were to sue (see below), but a reasonable argumet can be made in your favor.
HOWEVER, the at-fault driver and/or their insurer only has to pay *if* you sue, win, and get a judgment for the money. Anything they offer you without you winning a lawsuit is voluntary on their part, offered in an attempt to settle the case without litigation. What they offer is almost always less than you could potentially win at court: their incentive to settle is to save money, so if they were to pay you the full amount you might win, there's nothing in it for them--i.e. no reason to settle. 
So if you feel they are offering you too little, you have the right to sue for the full amount (full repair cost, lost wages, etc.) you believe you are owed. But lawsuits are never guaranteed--you can't be sure of winning, or how much you will win. And they take time, delaying payment to you. You need to decide if the additional money you want is worth the delay and uncertainty, or whether you'd be better off taking the proposed settlement.