"I remember speaking to my mum, she was emotional, she was in tears. You know she is just thinking 'Why him? Why him?' It's obviously not nice to hear, especially seeing your son getting abused," CNN quoted Abraham as saying.
"For me, I am always a strong character, it does not affect me as much. But saying that, it could affect people who do not have my personality. It was a challenging moment for me, I went through a lot of emotions. Everyone misses penalties but to miss a penalty I was obviously devastated," he added.
Over the past couple of months, several players have faced racial abuse on social media. Manchester United's Paul Pogba and Marcus Rashford were also subjected to racial abuse after missing goal-scoring opportunities.
Twitter had to intervene and they rolled out a statement saying "action against more than 700 incidents of abuse related to British football was taken".
"It just gives everyone an excuse to go online, behind their laptops, behind their phones, to say what they want. Some people might not think we see it, but we do see it and they just want to get a reaction. So Twitter needs to understand that," Abraham said.
"They talk a lot about cyberbullying and in a way that is, that is bullying over social media. I think some people just do not think footballers are humans, that they have personalities. We are humans, we do see it and it does affect us. Now Twitter needs to do something about it."
Before the start of the season, Abraham had played for Bristol City, Swansea City, and Aston Villa on loan basis for three years.
The 21-year-old credited the Blues' manager Frank Lampard for getting regular opportunities in the team and cement his place as an out and out striker.
"I had a lot of abuse but Frank Lampard always had his arm around my shoulder, lifted me up. Straight after, the boys, manager and everyone at Chelsea had my back. The next day, Lampard called me to find out how I was," Abraham said.
"He did not want to go over the situation because he knew how I was feeling, but just wanted to find out how I was, make sure that I was not distracted from football and my personal life was all right.It is nice to have that support behind you when you are going through times like that," he added.
Chelsea is currently placed at the 11th position in the Premier League standings and they will next take on Wolves FC on September 14.