Why can we breathe only oxygen?
You should have a look biological processes which are:


cellular respiration
photosynthesis


Loosly speaking, these two processes are just the opposite but deal with the same chemical compunds.

$C_6H_{12}O_6 + 6O_2 \xrightleftharpoons\  6CO_2 + 6H_2O (+ Energy)$

or in words...

glucose + Oxygen $\xrightleftharpoons\ $ Carbon Dioxide + Water (+ Energy)

From left to right is the process of respiration and from right to left is the process of photosynthesis.


Respiration releases energy from glucose (sugar). Therefore, animals, plants, fungi respire in order to get their energy from their food. More information on the chemical pathways on  Krebs cycle (or citric acid cycle). The main organelle involved in celullar respiration is the mitochondrion. Note: (Loosely speaking) the glucose is the end product of the digestion
photosynthesis allows organisms to create its own tissues from water and $CO_2$. But this cost energy. This is why plants use capture the energy form the sun with their leaves. More information on the chemical pathways on Calvin Cycle. The main organelle involved in photosynthesis is the chloroplast.


Note: In our lungs comes air which is a mixture of different gases (just like coke (Coca-Cola) is a mixture of different compounds). The two main compounds of air are Nitrogen (~78%) and Oxygen (~21%). We use only Oxygen. More info on the air composition here