We tend to translate "achzarius" as cruelty. The word conjures up pictures of brutality to others, and it is associated with the heartless mistreatment of animals. However, points out Rav Wolbe, from our Parsha we can glean an entirely different explanation of achzarius.
When Bnei Yisroel spoke disparagingly of Hashem and Moshe, Hashem sent snakes to bite Bnei Yisroel, and many people died as a result. Bnei Yisroel acknowledged their sin and begged Moshe to pray to Hashem on their behalf. Moshe acquiesced and prayed for them. Rashi comments that from here we see that if one's friend asks him for forgiveness, he should forgive him and not act with achzarius. We generally would not label a person who refuses to forgive someone who wronged him as an achzar; but Chazal do just that.
We find another two examples of achzarius mentioned by the Rambam. "One who fails to mourn a relative who passed away . . . has acted with achzarius. Rather, he should be worried and examine his actions and do teshuva" (Hil. Avel 3, 12). Additionally, "If the people do not cry out to Hashem [in response to a calamity that has befallen them], and they say it was caused by natural occurrences and happenstance; this is the way of achzarius" (Hil. Ta'anis 1, 3). Once again it is difficult to understand wherein lies the cruelty that classifies such people as "achzarim." So what does achzarius really mean?
Rav Wolbe (Olam HaYedidus pg. 20, 22) cites Rav Samson Rafael Hirsch zt"l who explains that "achzar" is really a compound word: ach (entirely) and zar (foreign). An achzar is one who acts entirely foreign to those around him. When a friend realizes the impropriety of his actions and wishes to regain his previous close relationship, he comes to ask for forgiveness. If the one who was wronged refuses to grant forgiveness, he has distanced himself and made himself entirely foreign to this friend. In a similar vein, one who fails to recognize that when tragedy strikes, Hashem is talking to him and trying to send him a message, has estranged himself from his Creator. Such a person lives in a world which is entirely foreign to Hashem's world.
Achzarius is the antithesis of what Yiddishkeit is all about. Judaism is supposed to bring one closer to his Creator and closer to the people around him. The Torah and mitzvos allow us to create a relationship with Hashem, genuine camaraderie with relatives and friends, and ultimately will bring true peace to the entire world.