Being the adverbial form of 余（あま）る ‘excess’, あまりに is a grammar structure that is used to convey an ‘excessive’ amount of something that caused a certain result. あまりに will almost always be used in sentences expressing some sort of surprise/shock about the degree of (A). Often, あまりに is translated simple as ‘so much (A) that (B)’, or ‘way too much (A)'.

• 今日（きょう）の仕事（しごと）はあまりに楽（らく）過（す）ぎて仕事（しごと）をした感（かん）じがしない。
  Work today was so easy that I don't feel like I worked at all.
• 今朝（けさ）は会社（かいしゃ）にあまりに遅（おそ）く着（つ）いたため、先輩（せんぱい）に怒（おこ）られた。
  I got to the company so late that I was yelled at by my senpai.
• 彼女（かのじょ）はあまりの恐怖（きょうふ）に、声（こえ）をあげて叫（さけ）んだ。
  She was so scared that she screamed out loud.

In the last example, we can see that あまり will be followed by の when linked to a noun. Occasionally, にも will be used instead of に. This just emphasizes the ‘excess’ of (A) further.

• 友達（ともだち）の話（はな）しがあまりにもつまらなくて途中（とちゅう）からほとんど聞（き）いてなかった。
  My friend's story was so boring that I stopped paying attention halfway.

As the literal meaning of あまりに is quite close to ‘excessively’, it may be easier to remember this translation, rather than ‘so much (A) that (B)’. Usually, ‘it is excessively (A), so (B)’ will cover most possible nuances.