When we interact with others, it's all too easy to forget that on the other end of that interaction is another live, sentient human being. Ego and the self can get in the way, which has the potential to distort the purpose in our interpersonal interactions. The reason that such שנאת חנם exists in the first place is that because people have a propensity to forget what it means to treat people with basic respect.
How we go about treating people is all the more pronounced during Tisha B'Av. Tisha B'Av is a time to mourn and grieve. Similar to the Exodus that we are supposed to relive during Passover, we are asked to relive the emotional tumult that the Jews experienced when they lost the Second Temple, or even when a subset of Jews was exiled from a given country. We are to recreate the sense of loss through memory. We realize that there is a time to feel suffering. Do we want fellow human beings, or even G-d, hating on us when we're down? Absolutely not! When we're down on our luck is when we need to be most consoled.
As Dr. Erica Brown points out in her well-written book In the Narrow Places: Daily Inspiration for the Three Weeks, there is one time in the year in the prayer book that we explicitly ask G-d to console us: Tisha B'Av. The additional supplication inserted in which we ask G-d to console us [specifically to rebuild Jerusalem] is נחם ("Console [us]"). While it's true that the Jewish people have restored Jerusalem back to Jewish sovereignty, there is still the need to relive what so many of our ancestors lived. Why? Because it helps us put ourselves in their shoes. It helps us understand the pain they felt, even though we never directly experienced it. It is a lesson in developing sympathy for others and the losses they feel. We should decidedly take the experience felt in נחם and apply it to our daily lives. How so?
We need consoling when things are not going well. We ask G-d to console us in our hour of need. If we ask this of G-d, we should also ask this of other human beings because we are created in G-d's Image. Suffering and struggle are part of the human experience. We all come across it a various moments in our lives. And when we do come across it, we hope that we can feel a sense of consolation. It helps to be able to reach out to others, verbalize our troubles, and have other people find ways to surmount the tough times. Even if they cannot do anything to change the circumstances, at least consoling you will help you weather it. As Dr. Brown brings up in her book (p. 75), "It is always more of a consolation if people understand our pain and reach out to us before we have to articulate our distress. We feel more loved when others can anticipate our feelings rather than when we have to spell them out. It makes us feel like we are the objects of their genuine concern. They have been thinking of us before we even told them of our distress."
In a more abstract sense, how does this work? The Sages had an answer to this: "Who is wise? He who foresees the consequences of his actions" (Tamid 32a). When we know someone is not doing well, we put ourselves in that person's shoes. The quintessential example of this in Jewish practice is when someone is sitting shiva. When attending a house of mourning, you should have the foresight, not to mention the social tact, to realize that there are certain things you do not say to someone in the grieving process. There are also proper ways to go about helping a person in their time of mourning.
It doesn't even have to be a time of mourning in which someone is in trouble. Take Abraham as an example. When the three men came to his tent, Abraham did not even need to ask what they needed (Genesis 18). Abraham had the foresight to realize that they were wandering in the desert, and that they would be hungry and thirsty. Without hesitation or question, Abraham immediately provided for all of their amenities because he had this foresight.
We can act with this sort of alacrity in our own lives. If someone lost their job, חס ושלום, you would let them vent, emotionally support them through their period of unemployment, and help that individual find a job. If they broke their leg or fell ill, חס ושלום, you would visit them on a bikkur cholim visit and do whatever you could to help ensure a speedy recovery. There are many opportunities that we can help other people, Jewish or not, to ease their pain and suffering, and in certain cases, even eliminate it. While consolation is hardly the only way in which we can connect to others, it is certainly a way that we can perform mitzvahs, help the world be a better place than it was before, and inculcate the appreciation of essential humanity of those who are around us. Only by reversing the שנאת חנם that destroyed of the Second Temple can we hope to bring about the Messianic Era.