Generations of students have been confused by the meaning of fictitious forces in non-inertial frames, but there is a very simple way to explain what a fictitious force is.
We start by reminding ourselves of Newton's first law:
Newton's First Law states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force.
Now suppose we lay out some coordinates with you at the origin and me standing still due North of you:
You observe me to remain at rest just as Newton's first law describes, so we know there isn't any force acting on me. Now suppose you start rotating anticlockwise. Now let's have a look at what you see:
In your (rotating) frame I no longer remain at rest, and indeed I'm not even in uniform motion in a straight line, so Newton's first law tells you that there must be a force acting on me to make me move in a circle.
And this is what we mean by a fictitious force. We know there can't be any forces acting because all that's changed is you have started rotating. But it looks to you as if there is some centripetal force that is pulling me towards you. This is the fictitious force.
Note that I've set up the situation so that I am stationary while it's you that is rotating. Typical descriptions of the situation involve stones being whirled around on strings while the observer in the centre is stationary. The trouble with this is that there are real forces acting, i.e. the tension in the string, and I think this makes it harder to understand what a fictitious force is. The centripetal force is real because it's the force the string exerts on the stone to pull it inwards. The centrifugal force is also real because it's the force the string exerts on the central pivot to pull it outwards.