Why did Batman design Robin's suit with only the underwear without pants?

Dick Grayson was an acrobat and entertainer. Thus his costume permitted the maximum flow of his body movements. He is wearing a leotard. It was Dick who molded his suit to that aesthetic.

New 52 depictions changed the origins and eliminated the tights all together. However the instance of its origins is still Dick's assemblage. He made it himself.

According to the creators, it was based on Robin Hood

The name "Robin the Boy Wonder" and the medieval look of the original costume were inspired by The Adventures of Robin Hood. {Jerry} Robinson noted he "came up with Robin Hood because The Adventures of Robin Hood were boyhood favorites of mine. I had been given a Robin Hood book illustrated by N. C. Wyeth ... and that's what I quickly sketched out when I suggested the name Robin Hood, which they seemed to like, and then showed them the costume. And if you look at it, it's Wyeth's costume, from my memory, because I didn't have the book to look at."

Source: The Comics Journal #271

The page shown above covers the origins of Robin, but not the costume or name, and the costume raised eyebrows from day one, so it was explained in a later issue. But Robin's costume, as well as his nom-de-guerre, come from a fictional (I think) organization called the Sea Robins. It was supposedly a marine-oriented boys group, like an ocean-going Boy Scouts, that Dick had belonged to. He liked the uniform so much that he designed his costume based on it.

So, it's not underwear; it's a Speedo-type swimsuit. And, as mentioned there are also tights involved.

Robin isn't just wearing underwear, he is wearing a leotard. Dick was part of a circus and his mum and dad were the trapeze act which he was also training in. After his mum and dad are killed in their act because of local mobster Tony Zucco taking revenge on the circus owner, Batman finds Dick. Dick convinces him to let him help and becomes Robin the Boy Wonder. His outfit consists of his leotard because of his circus background. As such he is essentially wearing clothes he is comfortable in for the role he will perform. This all happens in the first few pages of Detective Comics #38.

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Of course it's worth noting that over the years there have been different Robins and different costumes. Most of the newer ones include trousers/tights in them.