(13)The more he leaned the more it bent, until he could almost reach the honey.
「lean」「bend」については解説済み。

「reach」は、日本語で「リーチが長い」というように、語源は「手を伸ばす」です。現代では「対象・目標に届く」イメージで使われます。「に着く」「に届く」「に手が届く」「手を伸ばす」のような意味で使われます。
We reached the hotel at midnight. (got to / arrived atの方が口語的)
His e-mail reached me just now. (I received/got his e-mail just now.)
Can you reach the key on the top shelf?
She reached out for the book.(彼女は本を取ろうと手を伸ばした)

(14)He leaned just a little bit further when … CRACK! The branch snapped.
「lean」については解説済み。

「snap」は「ボキッと折れる」「プツンと切れる」の意。「break」でも伝わります。
The branch snapped when he hung his bag on it.
The rope snapped.

(15)Pooh bounced from branch to branch, then BUMP! He landed headfirst in a bush!
「bounce」は「弾む」「弾ませる」の意。「rebound」でも同じ。
The ball bounced over the net.(ボールははねてネットを越えた)
She was bouncing a ball on the sidewalk.

「land」の語源は「陸」。動詞では「上陸・着陸する」「上陸・着陸させる」の意。反対は「take off」。「touch down」でも同じ。
We are going to land at Narita in thirty minutes.
The boy sent flying into the air and landed 20 feet away.