According to the Talmud, the reason for the deaths of Rabbi Akiva's students was "shelo nahagu kavod zeh la'zeh" (Yevamot 62b) - they did not treat each other with respect. This may seem rather trivial at first glance, but a further analysis explains why a lack of respect for other people has the power to destroy. We all cherish our own opinions. Whether we believe in a certain approach to religion or politics, we love to have our voices heard. But if we are to remain a civilized people who value the truth, we cannot simply demonize our opponents. While there is nothing wrong with standing up for a strongly held belief, there is a great deal wrong with being disrespectful to ideological adversaries. Instead of a vigorous intellectual debate, our differences will turn into a game of whose ego prevails. And instead of a war of ideas, it will become a war of personal attacks.
A good example of how we can benefit from being tolerant comes from the many debates between Beit Hillel and Beit Shammai (the Schools of Hillel and Shammai). The Talmud (Eruvin 13b) explains that we follow the opinions of Beit Hillel because they always studied their own rulings along with those of Beit Shammai, and even mentioned the teachings of Beit Shammai before their own. On the other hand, Beit Shammai was so certain they had the whole truth that they didn't bother to study the teachings of others. Since Beit Hillel showed tremendous humility and tolerance, they were not only ethically worthy of being chosen over Beit Shammai, but also more likely to reach accurate conclusions in their studies. Beit Hillel always had to defend and refine their own views.
There is a concept brought down in Pirkei Avot (3:17) by Rabbi Akiva that states "syag la'chachma shtikah" - a protective fence for wisdom is silence. This phrase doesn't mean that we have to remain quiet in every conversation or that we cannot object to someone's ideas in an argument. It simply means that if we constantly speak over others during a debate, we will never have the chance to gain wisdom. We learn the most when we listen - not when we talk. Therefore, we will neither grow interpersonally nor intellectually when we are readily dismissive of other people.
During this period of semi-mourning for the students of Rabbi Akiva, are we just going through the motions or actually trying to solve the problem that led to their downfall? Perhaps it was after this tragedy befell his students that Rabbi Akiva made his famous declaration, "to love your neighbor as yourself is the major principle of the Torah." Of course, becoming a better person is always easier said than done. But if we get in the habit of politely listening and calmly responding when we disagree with someone - instead of rudely interrupting and lashing out against them - we will not only become more respectful, but also wiser in the process.