so i was studying LM386 datasheet today.

check the attached pic.

one circuit "amp1_cropped.png" is made without any use of capacitor and another circuit "amp2_cropped.png" is made using capacitors. The output are same. So why designers use capacitor in the circuit. 'And how to calculate the capacitance of those capacitors.

 

I think the 250uF polarised capacitor is there to block DC offset that could damage your speaker.

It might not matter so much with the simple LM386 amplifier but electrolytics are generally considered bad for audio, but considering that it's 250uF required then it's pretty much electrolytics that you are stuck with.

In your simulation the output is the same because it will probably simulate the capacitor as being perfect, while in the real world it wont be as it'll have all sorts of distortions and imperfections.