Carol Monaghan was moved to a safe house after online abuse turned into a detailed threat.
Earlier this month Jonathan Bell, 35, admitted his behaviour caused Monaghan “fear or alarm”.
Monaghan is now calling for more protection for women, saying social media companies must take action to end misogynistic abuse.
MP’s office was targeted
Speaking to BBC Scotland’s The Seven programme, the MP described the messages taking a sinister turn, with one including references to the murdered MP Jo Cox.
Her constituency office in Partick was targeted, with windows being smashed. The office front was also splattered with ketchup.
"It was obviously meant to look like blood across the windows. That was the start of the physical activities."
A death threat was then made against the MP.
She said: "It was phoned in and it contained enough details about my personal life, enough detail to cause the police to take it seriously," she explained.
"I got a call from my office manager. The police had contacted him to say there was what they considered to be a credible threat.
"They weren't necessarily sure I should come back to Glasgow. But I was keen that I did come back to Glasgow - my family were here."
Terrified for her family, Monaghan added: Because of the personal nature of the threat and the personal details, I knew he knew where I lived, I knew he knew who my kids were. I just had to be there.
"That evening when we came back from the police station to the house, the police wouldn't let us into the house until the whole area had been searched.
"They spent the night outside the house and very early the next morning we left and went to a safe place."
Social media companies must take action
Monaghan has called for social media companies to take responsibility for the abuse many suffer.
She said: "Social media gives people a platform, it gives them a way of directly contacting a person - at any time of the day or night.
"It gives them the opportunity to say things anonymously. It gives them a whole lot of protection that the target of their abuse does not have."
Monaghan added: "Really, why should we take that level of abuse? We wouldn't accept it in a workplace, but we are just supposed to take it and somehow we are to blame if we don't.
"I understand some people want to interact on Twitter anonymously for the best of reasons but there is also a huge hiding place for people who want to put more sinister stuff out there. We need a way of identifying users and take action against them for tweets that are offensive."