I expect the reasoning behind it is to make people stop and think can I afford more than two children
No the reason behind it is to keep us all squabbling amongst ourselves so we don't notice where the money is really going, and it's working very well.
The 2 child benefit policy is obscene and will save no money whatsoever.
It's obscene because -
It says some children are not worthy of support. There will be children born third or later into poor families who we as a society have decided are not worthy of supporting. There are no 'better decisions' those children could have made. Nobody has a choice about the birth order or income of the family they are born into. We are deliberately withdrawing support from some of the poorest children in the country. We are punishing them for an accident of birth.
It will prevent lone parents on low incomes from forming families and living with the people they love. If you are dependent on CTC and have 2 children, and your partner also has a child, you will have to think really hard about whether you can afford to move in together and form a family. The right to family life is part of the human rights act. It is fundamentally wrong to keep families apart and you need a bloody good reason for doing it. Being poor is not a good reason.
It will force women who have been raped to jump through hoops and justify themselves to get support for their children. They will have to declare on a form that they have been raped or coerced and get a third party agency to verify that they are not lying. This is humiliating shit that no man will ever ever be faced with. Every time a woman has to show her TC award letter to someone (not infrequently, if you're poor) people will know. If a woman still lives with her abuser and hasn't yet managed to escape then tough shit, her child is not worthy of support. Leaving an abusive partner is a very dangerous time. I expect there will be more women who are murdered while trying to leave abusive men because they can't afford to support their children under the new rules.
It reinforces the mother / whore dichotomy. If you enjoyed the sex then no, your third child will not be supported, whatever good reasons you have for accidentally procreating. Shame on you! If you were raped or coerced then OK, we'll support your child as long as you provide frequent documentary evidence of not enjoying the sex and have this verified by someone who knows better than you.
Weird how there are people who oppose abortion but make an exception for rape.
And there are those who oppose supporting lone parents, unless their children are a product of rape.
And I suspect there is a fairly large overlap between those two groups. It's almost as if this is not about the children at all 
It won't save any money because -
It's an administrative nightmare. DWP are having to come up with processes on the fly to comply with hasty, rapidly changing legislation. This has massive cost.
There will be additional costs for DHP's, social services support, extra NHS costs. These won't show up on DWP budget so expect a few good news stories but the costs are still there and we are all still paying.
Another big cost is staff turnover. The people who are expected to administer this system are underpaid, undersupported and undertrained. Lots of them give up, even if they went into their jobs with the best of intentions, and the more complex and inhumane we make the rules the more dedicated employees we will lose.