If my "price per share" is same as "exercise price", do I need to file?
Yes, you should file a 83(b). Otherwise, you'll be paying taxes every time the stock price increases.
If you don't file, you pay taxes every time you vest. If you do file, you pay those taxes upfront. These upfront taxes will be be trivial because the stock isn't worth much then.
If your stock is $1 today and increases in value $1 every month, you're now paying more taxes every single month. Instead, if you file an 83(b) you pay taxes on the $1.
I thought I pay tax only on two occasion: when I exercise and when I sell stock. Do I need to pay taxes by just holding it?
If your stock rises in price, you need to pay tax for the capital gains you're getting from the increase in the stock price, rather than just paying no tax for the stock as it rises in value – because you paid it all at the beginning when you filed the 83(b).
This is why you should file your 83(b) election after exercising options.