I think Obama is somewhat conflicted over the balance and is unable to reveal the interesting parts of that question.
Assuming an environment where all digital information is monitored and it actually is used in a reasonable way (active, investigations, internal warrants from courts, etc.) I could see there being an argument that keeping the extent of it secret prevents people from knowing what's being monitored and avoiding those methods of communications. I don't agree with this argument primarily because things change in leadership and governments and having a system like this (especially with it being secret) can lead to abuse.
This was Snowden's point about changes in policy and 'turn key tyranny'.
Clearly access to this information changed Obama's opinions pretty severely - there must be a lot we don't know.
What if it had prevented a biological weapon that could have killed millions of people in NYC? Revealing this publicly could cause mass panic.
Obviously I have no evidence of something like this, but it's a thought experiment to consider. Would that make the surveillance worth it in your mind?
The thing is, I don't think anyone is claiming major successes. They're just saying that it's OK. Watching Obama's talk just gives you the feeling it's political nonsense, with a bit of CYA lines too.
In the case of such a broad surveillance system, I'm having trouble coming up with good secret reasons. Perhaps I'm being unimaginative. If, for example, this system was preventing hundreds of terrorists, what damage would be done by stating that it has unequivocally stopped major threats?
What's the damage in revealing what's going on and why?
Is the plan "if we're quiet enough, bad guys won't bother using encryption or cover their tracks?
I'm curious to know, what information, in theory, would change someone's opinion, and must remain secret in order to keep things safe. Aside from breaks on popular ciphers or mass collusion with major hardware/software vendors.
Edit: I'd trust a president stating for sure that this is entirely necessary, and stating there is secret information that'd even change the EFF's opinion. Obama's speech doesn't give that perception at all. It feels like he's stumbling around to justify something he knows is bad, using nebulous threats.