Ok, the picture makes it very clear.

It means to clean up after your pet's droppings (as Jungkook Kim said), as part of a cleanup campaign.

Instead of saying "Take care of your pet's poop" to the owner, it has the pet saying "Master, please take care of my poop".

It's a cute way of getting the message across, as is popular with many such campaigns in Korea and elsewhere.

(I was not imaginative enough to figure that out without seeing the picture, haha)

It's similar. Since 응가  means poo, 응가를 책임져 주세요 is 'Please, take care of me.'

In this sentence, Korean can imagine that animal might envy other animal because of the word '응가도'.

As '도' means too, in this situation, exact meaning is 'Master Please, take care of me, too'.

i saw this sentences when i traveled korea times....

https://www.dropbox.com/s/wv7vpyqqqtyluw4/20151111_151157.jpg?dl=0 ;

That sentence is really weird for native Korean. However, I can imagine what the sentence means. I guess it is some kinds of a campaign; like "Don't mess with Texas".

Basically, 응가 means poo, and sentence says that "Master, please take care of my poop too". This means you should not only take care of your animal but also take care of your animal's poop. In other words, it means don't leave your animal's poop behind. 

That is a weird sentence.  Can you imagine saying "Master, please take care of my poop too".

That is the direct translation of it using the most common meaning of 응가, which sounds too weird for me.

However, 응가 is a baby word (under five or so), and it's neither universal (some say 응꼬) nor offensive.  

So it might be someone's name too - not likely in real life but easily possible in storybooks and cartoons.

In some parts of Korea, 응가 is a dialect word for an older sister, so there is that slight possibility too.

Anyway, if you really want to know what the sentence means, you should consider the context.

Try and see if it can be poop, an older sister, or a name of a character :-)