Question

why do we breath fast after running?

Answer 1

because anaerobic respiration takes place

Answer 2

Not an Biology Expert, But answering What I know, Here it is,

Please read, The last 3rd Para If you need the answer in short !

To make our Body alive and Stable , We have to respire every second, By respiration we take , Oxygen , Which is very useful to our body, This oxygen is taken into the blood, and is transported to every part ,

I told you this, As this is important in further topic,

Now, For every exercise and Work, We need Energy,
We get energy by breaking down the glucose, and converting it into A.T.P, 

For the exercise we need to use More A.T.P, For running it should be used in more Quantity, A.T.P Stores a maximum of 7200 Calories Energy,

So, for running we need more than this, So, How can we get it?
There are two types of Converting Food into Energy, 
(1) Anaerobic Respiration (Oxygen is insufficient)
(2) Aerobic Respiration (Oxygen is Sufficient),

For Converting Food into ,Energy, First, The food is broken down and is converted into Pyruvic acid, If oxygen is present, Aerobic Respiration takes place, 

And the products we get it is,
CO2 + Energy + Water (H2O),

But If we have less oxygen than required after converting food into Pyrivic Acid, Then Anaerobic Respiration takes place, 
And the End products can be,

(1) CO2 + Energy + Lactic Acid,
(2) CO2 + Energy + Ethene ,

Now, For running , Every person has to breath more than usual first and then race without breathing at all, 

After the race, every person will feel different , That is, Everyone breaths hardly to take in oxygen,

What happens behind this, Anaerobic Respiration, Here the 1st case takes place, Due to insufficient Oxygen, We produce more lactic acid, and more energy , But energy is less when compared to aerobic respiration,

But, When we are running, We won't have time to take in oxygen, So after the race, More lactic acid is formed, and to suppress Lactic acid and To make body stable and active again without Lactic acid, We breath hardly and take more Oxygen at faster rate, i.e We breath Fast,

So this is the reason for your Question,

Hope you understand, Have a Great day !
Thanking you, Bunti 360 !