I'm afraid the claim that they will definitely save you money is false. They can save you money if your previous bills have been estimating your usage as too high. But if your bills have been estimating too low, then the smart meter will more accurately bill you and your bills will go up.
So it depends how often you normally send your meter readings in. If you send regular readings then you probably won't see much change. If your bills are estimates, then it could go up or down.
Likewise, they also enable you to see the cost of your energy in real-time, so the theory is that you'll see how much using X Y Z appliance costs you at peak times and then in theory you can change your habits to conserve your energy use. In practice (my personal view having had a smart meter for a year), it is unlikely to make a real difference if you're not ridiculously excessive in the first place. But it could help if you're struggling with high energy bills and want to identify where you can cut your costs a little.
But thinking more broadly than this, smart meters are pretty important for the future of the grid as we turn to more reliance on renewables and battery technologies etc. Obviously back in the day if demand was high we could just switch the coal fired power station on, now we can't just generate more wind out of nowhere. So storage is important and understanding demand is also important. Smart meters are a part of this move to a more "smart grid" approach where the balance of supply and demand can be monitored more effectively. Longer term, when we have things like electric vehicles and our own renewable energy sources at home, it'll all form part of a more dynamic grid system where we as consumers can play a part in balancing the supply and demand (for example being paid to generate energy and put it back into the system in times of high demand).