It's about principle. As she is Ted's ex-girlfriend, Ted can't tell Robin who she can and can't sleep with. But he can ask a bro to back off his ex. However, he wasn't as mad at the fact that he slept with Robin as much as that Barney's thing is that while he's a horrible person, he's a good friend. Ted has stood by Barney for years, watching him do terrible things to other people, but always thought that Barney had a limit - that he wouldn't do such things to his own friends. So when Ted finds out that Barney broke the Bro Code, which Barney constantly promotes and Ted in turn bends over backwards to uphold, he was understandably pissed.

I guess you're right. Barney certainly made it clear in the most recent episode that he lacked principle - deciding that Nora wasn't the right woman for him, after cheating on her with Robin. What a prick!

Ted had no right to be mad at Barney, in an earlier episode we see Barney and Robin at lazer tag and Barney calls ted who was dating...someone, I forget who, else and Barney asked ted if he could go after Robin. Ted gave Barney the go ahead. He has no right to be like "how dare you sleep with Robin" when he earlier said "go ahead try for Robin good luck with that"