Why should the buyer of the bond to pay for interest accrued to the date of issuance?
Why company just not take par from the buyer and pay him interest for the remaining period?

There are certain things that are traditionally done this way.

First of all, you can purchase a bond from someone else, not just the company. It's just like buying stocks -- unless it's an original issue, you're buying from someone else, not from the company. From this point of view, it makes a lot of sense for the buyer to pay the portion of the interest to the seller that belongs to the seller. The seller held it for that period. The company will pay the entire six-months to whomever owns it at the time. If you have it 4 months and sell it to me and I have it 2 months, I will get all of the interest. So I pay you for 4 months up front, then I get all 6.

Property taxes on homes are generally done this way also. The buyer has to pay the entire thing when it comes due, so the seller pays them in advance for the portion the buyer is going to get charged before they bought the house.

In the case of original issue direct from the company, it's still for the sake of ease. They could be selling bonds at all sorts of different dates. Then when interest is due, they'd have to go through and figure out how much each bondholder is due for that period. It's a lot easier for the buyer of the bond to pay the first portion up front to the company, and then when the company pays interest to everyone, they can simply pay six months to everyone and not have to figure them all individually.

In other words, it really is actually easier that way. If you're having difficulty understanding how the math all works out, that may be hard to believe. But once you understand how it all works, it really is just easier that way. (At least to the company - perhaps not to the buyer.)