You are right to feel empathy about the situation, and it is good to express empathy to our patients. I applaud you for not letting that get beaten out of you by the difficult training our system puts us through. Keeping the human element in medicine is important not only to our patients but to ourselves and the culture of medicine.
And you are right to seek guidance. GME offices usually prep residents in orientation for how to approach this; it is unfortunate that yours did not. It is important to remember that what you say really does matter, both psychologically and legally.
Talk with your faculty about how they recommend you approach it, and I would specifically seek out the attending with whom you precepted the patient. Remember, it was their patient too. If you do not feel comfortable talking with that attending because of their personality or attitude, turn to your residency director or faculty advisor - that's what they are there for. There is also going to be a legal consultant for the hospital, and you can ask them advice as well.
But one important distinction is whether you truly made an error, or whether you are experiencing empathy about an unfortunate case of an atypical presentation falling through the cracks of standards of practice. Both are types of errors, but they are distinct, and have very different implications. I'm assuming you feel like you missed diagnosing the SAH when they were under your care, but I also assume that you practiced medicine using usual standards of practice while they were in the ED - e.g. following standard algorithms for when to order head CT after a fall, etc. The sensitivity of those tools are not 100%, but they are often the best tool we have when weighing the risks associated with blasting someone's brain with CT radiation against the likelihood that this particular headache is actually a brain bleed. At a population level, think of the brain cancers from unnecessary CTs these algorithms are preventing by not scanning every headache or bumped head. The decision for an individual case is not easy, which is why these algorithms exist. Consider what really happened, not just with the knowledge of the ultimate result. Hindsight is always 20/20.
And again, don't do this on your own. Plus you can recommend to your residency director to hold a didactics session on communicating medical errors; he/she could invite the legal team to discuss it with the residents. It is an important professionalism topic that should be part of your training.
Some websites with good reading on how to approach discussing errors: