How exactly did 24,000 of Rabbi Akiva’s talmidim die from not being respectful to each other? Were they killed by the Romans? Did they develop a disease? Is this discussed in the Gemara & where?

The source of the story is Yebamot 62b and medrashim. Some sources mention that the talmidim died of a plague of diptheria. As with all “aggadta”- the account should be studied so as to find its deeper meaning. The Avnei Nezer (Neot HaDeshe) writes that the talmidim of Rabbi Akiva were so committed to the broad national idea of building Torah and restoring Jewish independence that they lost sight of the unique contribution made by each of them individually. This is why they didn’t respect each other as individuals. They were absorbed in the bigger picture and this is wrong. One has to see the bigger and the smaller pictures at the same time.