An expression indicating that something was on the verge of happening — 'almost' or 'just about to'. Similar to 'もう少しで'.
もう少しのところで転びそうになったが、手すりにつかまって助かった。
I was about to fall, but I grabbed the handrail and was saved.
プレゼンで緊張して、もう少しのところで言葉が出てこなくなった。
I got nervous during the presentation and almost couldn't get the words out.
もう少しのところで優勝を逃したが、最後までよく頑張った。
We just missed winning, but everyone worked hard until the end.