At first, he didn't want to show us his picture, but after we kept asking, he reluctantly showed us.
「最初彼は写真を見せるのを嫌がったが、何度も頼んだら渋々私達に見せてくれた」はちょっと長い文章なので、半分ずつ説明します。
「最初彼は写真を見せるのを嫌がったが」の部分は英語で：
At first he didn't want to show us his picture
Af first he didn't want to show us his photo
At first = 最初
he = 彼
picture, photo = 写真
show us = 見せる
didn't want to = 嫌がった
「何度も頼んだら渋々私達に見せてくれた」の部分はこのようになります：
after we begged him many times, he reluctantly showed us
after we kept bothering him, he unwillingly showed us
after we kept asking him, he reluctantly showed us
many times = 何度も
begged him ＝ 頼むの意味ですが、ちょっと普通の頼むよりもっと強調します。
kept bothering him = 迷惑をかけるほど頼む時に使えます
kept asking = 何度も聞くこと、何度も頼むこと
unwillingly, reluctantly = 渋々
showed us = 見せてくれたら
完成した文章はこのようです：
At first, he didn't want to show us his photo, but after we kept bothering him, he unwillingly showed us.
At first, he didn't want to show us his picture, but after we kept asking, he reluctantly showed us.
At first, he didn't want to show us pictures of him in costume, but we wore him down and he reluctantly showed them to us.
He didn't want to show us his pictures, but we pestered him until he gave in and unwillingly showed them to us.
We kept asking him to show us his pictures until he reluctantly did so.
(1)のwear someone downは「疲れて納得するまで説得する」という意味です。ちなみに、She was worn down.なら、「彼女は疲れ果てた」という意味になります。
「 渋々」はreluctantly、「嫌々」ならunwillinglyなどの副詞を使います。
「何度も頼む」は、(2)のpester, (3)のkeep asking, ask repeatedly, ask over and over againなどで表せますよ。
アメリカ合衆国 
