The 27-year-old took to the stand at the High Court in Pretoria on day two of his testimony during a trial in which he is accused of murdering Steenkamp on Valentine's Day last year.
The double amputee told the court that his relationship with the South African model "got going" in mid-January after they first met in November 2012 at the Sports Awards.
While in the witness stand, Pistorius added: "I was very keen on Reeva, I think if anything I was maybe more into her than she was at times with me, and I let her just take her space. I was just besotted with her."
Pistorius denies murder by maintaining that he mistook Steenkamp for an intruder when he fired gunshots through a bathroom door at his home in the early hours of February 14, 2013.