Well, they don't have to, since that's hopefully covered in your undergrad days, but the president seemed to have missed a few math classes along the way.
"These so-called right-to-work laws, they don't have anything to do with economics; they have everything to do with politics," President Obama said.
"What they're really talking about is giving you the right to work for less money."
Now, if I remember the law correctly, you have the option to not have dues taken out, which would give you more money than you had before the law passed. The only way this would create less money for workers is if the union paid you to belong, supplementing your income.
The liberal logic here is that it would make it harder to get raises if not everyone was paying dues; also, if only 80% or so of folks went on strike, the 20% of non-members would make a viable nucleolus to work scab-temps around, thus making a strike less painful to management and less useful as a bargaining tool. If that hold, it would create downward pressure on wages; it's an open question whether that effect will be less than the union dues.
However, in the short-term, it's the status quo for the union members and a modest raise in take-home pay for the dissenters. It's a modest drop in take-home pay for union HQs; that's why they doth protest too much.