It's a very good question.
The first thing to recognise is that modern Western society runs on very aggressive, competitive principles. Phrases like: 'dog eat dog', 'winner takes it all', and 'no time for losers' kind of sum up the general attitude that's so prevalent in 2015.
We're taught from a very young age that life is a competition we have to win, that we can 'make things happen', that all success in life just boils down to our own effort, hard work and abilities. And of course, there's some truth in all of this, otherwise it wouldn't stick.
But reality is much more complicated than the aggressive competition paradigm likes to pretend. The reality is that there are plenty of hard-working, accomplished people in the world who put lots of effort into things, and still don't succeed. There are also very unaccomplished, not very hard-working people that often achieve status and success that's far beyond what their modest talents and efforts would have predicted for them.
Why is this?
We're not in control
Simply, we're not in control of the world, and it's not all 'down to us' to stay healthy, or to get rich, or to be thin, or to have the perfect home, or whatever the particular goal happens to be.
The more we realise that, the more compassion we have for other people, because we start to understand that our successes and triumphs in life are all presents. If God decided that we were going to lose all our money, hit the skids, and turn into the world's biggest 'loser' tomorrow, that's EXACTLY what would happen.
The more we understand that, the less arrogant we are about our successes, and the more compassionate we are to people who are losing or failing in life in some way, because 'there but for the grace of God, go I.' It only takes one piece of bad luck for people to lose their homes, fall into debt, go into receivership, lose their health and be unable to work, for the tables to turn, and for us to see how limited we human beings actually are, all their previous 'successes' notwithstanding.
The first reason people see compassion as a weakness:
So the first reason people see compassion as a weakness is because they've bought into the paradigm that cuts God and Providence completely out the picture, and says that success depends only on a person's own hard work and effort. So if someone isn't 'successful', you can easily rationalise that it's because they're lazy / stupid / nasty or somehow not deserving of success, which makes it much easier to put the boot in, instead of feeling empathy for them.
The second reason people see compassion as a weakness:
The second reason that people see compassion as a weakness is because it impedes competition and 'winning', if you start worrying too much about how the other guy's doing.
In the rat race, you're not going to stop to check on the guy who's hobbling at the back of the pack, because that's going to slow you down, and prevent you from coming first. In the competition paradigm, everybody else's loss is your gain, and you'd have to be an idiot to through away your own competitive edge by wasting time worrying about other people.
Of course, taken to extremes, this lack of compassion for others is the basis of pretty much every mental illness and criminal behaviour knows to man. But while most of us like to think we wouldn't stoop to mugging nannies, we don't realise that aggressively 'looking after number one' when it comes to our competitors, work colleagues, friends or relatives is actually rooted in the same bad character trait, of lack of compassion.
The last reason people see compassion as a weakness:
The last thing to say is that compassion is not the norm in today's world, so as well as not treating others with compassion, we are also not treating ourselves with sufficient, healthy, self-compassion, with some huge consequences for our own peace of mind, happiness and quality of life. If we can't treat ourselves compassionately, we certainly can't treat others (including our nearest and dearest) in that way.
There's much, much to say about this, but I'll stop here for now and point you in the direction of a couple of other interesting resources:
1) Visit the website of mental health researcher Kristin Neff, who's done a lot of work on the huge physical and mental health benefits of practising and improving your self-compassion.
2) Visit the Jewish Emotional Health Institute website (JEMI), where you'll find a lot of interesting information about why compassion is lacking, why we so badly need to develop more of it in order to live happy, healthy lives, and how to start going about doing that.
3) Take a look at this article about the energetic dimension of compassion. Energetically, spleen meridian is linked to compassion. As well as being responsible for happiness and metabolism, weak spleen meridian energy also plays a huge factor in allergies, immune function, learning difficulties, and I believe it's underneath a lot of what's happening with things like ADHD and personality disorders, too.
The more compassion a person gives and recieves, the stronger their spleen energy gets, and the less problems they have with allergies, learning difficulties and a whole bunch of other things that are SO prevalent in 2015.