How does a monk cope with all the bad things happening in the world if the only thing monks do is sitting and meditating? Don't monks have any urge to be an activist against for example all the violence against animals and human rights where they can help other people for their needs. How do they deal with those feelings of injustice, how can they justify themselves not to declare or demonstrate or be active in other way to stop these cruel things if they have a heart for the world. How can they just sit and meditate in a monestary?
This question is more broad than monks and monasteries.  Just because they focus solely on meditation does not mean that everyone else is involved in addressing injustices... so, let me just slightly change your question to "how can one sit and meditate while..."

Where does all the suffering in the world come from? 

Anyone who inflicted any suffering on a human being or an animal - wasn't it something in their minds that caused them to do that? Isn't the addressing of those causes in the minds critical to the lessening of suffering?

From the perspective of the one suffering - one can see quite quickly that the mind plays a huge role in interpreting events that happen. The sole difference between a painful event and something that happened and got taken in the stride, is the mind. 

These two aspects taken together, argue strongly for mind training as the single biggest tool to counter suffering. 

But...one may ask "How is working on my own mind helping anything? A is causing suffering to B. If anything, it's their minds that should be trained, not mine. My job is to fight A and save B".

Anyone who sits in meditation knows pretty soon that one has little control over one's own mind, let alone on anyone else's.  At least, with one's own, there is some chance of influencing it. And why does that matter? Because we ourselves become A or B in situation after situation, many times a day (if we are mindful enough to notice that). Hard as it is to admit that I am an inflictor, it is easy enough to see that on the other hand, every inflictor is an "I".  It is a fallacy to think that some people are of a different type (evil) and the rest of us are not. It is just the same type of mind, having been conditioned by different circumstances, which acts out differently.  It is rare that one does harm for the sake of it. It is far more common that one just seeks what one desires and acts in a way that doesn't care about what happens to anyone else because of that.  That includes some of the most heinous crimes in history like slavery and various kinds of exploitation happening up until today.  

Are we aware of how, in the pursuit of convenience, fun and pleasure we ourselves cause harm without meaning to? Plantation owners in the American South and farmers under apartheid were just "trying to work the land..." right? The brutalities committed in Africa and elsewhere were just to "help industries back home in Europe by finding them resources", right? Or perhaps it was even the noble cause of "scientific and technological advancement of society back home".  Until recently, we thought we were just driving to and from everywhere, having fun and convenience, enjoying the freedom of the road, not hurting anyone at all until having to face the catastrophic effect on the climate. Think my choices of buying and consuming something has no effect on human and animal suffering? I need to think again! So the mind, even of "us innocents who aren't hurting anyone" could use a lot of examination and cultivation. 

This is not to say that one should not do anything externally to help anything. By all means. But while doing so, no need to come to it from a place of anger and agitation towards anyone. We can work towards goals but not neglect what is happening with the mind. 

Also not to say that protest and activism don’t have their place, but it is a huge mistake to think that focusing on the mind is not one of the most powerful tools at our disposal. If you don't believe that, try this experiment - every time we ask "how can I deal with this person at work/school who is being a _____?" Also ask yourself "how can I make sure I'm not myself being like that person?", see what in the mind is causing such behavior from oneself.  And see the magical effect this has on the environment around you after a period of time.  Quite a different thing from taking refuge in identity ("they" are bad ones) instead of seeing what one can do on one's own end.

There is also something odd about asking this about monks and meditation alone - as if it is the sole alternative to human rights work or even remotely a factor in the causing of suffering? One can ask the same about a whole lot of other things - "how can someone watch TV while so many people are suffering?". Or going to the movies, or a bar or studying mathematics or philosophy, doing photography, sleeping in on a weekend, going on a trip with friends, playing with your children.. do people do all this because they "don't have a heart"? 

As for monks...every field benefits from people who do it full time. Monks are those "pros" in the context of cultivating the mind. So if you can grok the value of mind work, such a question would not arise about monks. 

Finally, I might add, it was precisely with the goal of reducing human suffering that the Buddha chose to go forth and search, and then to teach the path he found to others - not in disregard of it.