Alfred leaves Bruce, telling him that he has buried enough members of the Wayne family. But, Bruce has been running around all this time as Batman. I think the odds of him being killed are pretty high. Why is Alfred only worried now?
When Bruce Wayne returns as Batman in Dark Knight Rises he isn't the same person in terms of his reaction times and agility which he possessed years ago when he decided to hang up his boots at the end of Dark Knight (plus the limping at the start of the movie). Alfred strongly believes that he is also inferior in physicality, strength and reaction to Bane when he sees all the footage of him during the motorcycle chase scene he enacts in front of Gotham.
Alfred tries to explain the scenario to Bruce Wayne to the extent of Bane's strength and also believes that Bruce Wayne is trying to be batman to satisfy his obsession, even though he is old Bruce Wayne still has the heart of the old batman and tries to counter him stating that his weakness can be exploited which ultimately leads Alfred to leave on the fact of their heated discussion.
All those years Alfred never gave up on Bruce because he believed that he could fight those odds but in the case of Bane, Alfred instinctively knew that this was the fight Batman can never win at that point of time given his strength and also earlier Alfred hints to him (the coffee place he talks about) about starting a family (or finding the right girl) which could have led him to that decision.
(Alfred is correct at that point and Bruce Wayne gets his broken back immediately after he leaves)
Alfred burnt the letter from Rachel that stated she'd chosen Dent over Wayne. This obviously puts a strain on their relationship so he leaves.
Alfred, in the stairwell, mentions something along the lines of: "leaving is all I have left...". He is trying to get Bruce to stop being Batman, as others have said, because he believes Bruce is too old and not strong enough to survive the life anymore. He tries to talk Bruce out of getting back in the Cowl, then tries to scare him out of it by telling him of Bane's legend, he even confesses the truth about Rachel to get Bruce to move on from his past. When Bruce still won't stop, Alfred desperately tries the one thing he hasn't tried yet, leaving. He doesn't want to do this, because it will make Bruce hate him and it will make him lose Bruce. "I know what this means, it means your hatred, and me losing someone I've cared for since I first heard his cries echo through these very halls"(this is me paraphrasing from memory as best as I can). He follows this ith saying essentially that if he does indeed leave, maybe it will make Bruce see that he should quit.
The idea is that Batman and Alfred's relationship is based on an almost parental love and Alfred wants to show that his love is now leading him to remove his support and declare his disapproval even to the extent mentioned above.