If centrifugal force isn't real then why am I being pushed on a merry go rournd?
It is not you who are accelerating outwards away from the car seat or the merry-go-round - it is the car seat or merry-go-round that accelerates away underneath you.
And as it moves away underneath you, the car's door or merry-go-round's rail soon bumps into you and pulls you along. This constant inwards pull is what makes you move around the circle.
- Think of you pulling the duvet away from underneath the plates, glasses and cutlery on a small dinner table. Everything stays where it is, but the duvet moves away.
From the perspective of the duvet (if you placed a camera on the duvet), though, it looks like it is the cutlery, plates and glasses that are moving away. But we know that isn't true; that is just an illusion. If you were that glass and all you could see around you was the duvet, then your brain would easily trick you into thinking that you are the one moving rather than the duvet. It is a typical optical illusion.
- Another example: imagine standing in a bus while wearing roller skates. When the bus brakes and decelerates, you feel like falling/rolling forwards. But nothing is pushing you forwards. It is rather the bus around you and underneath you, which is being pushed backwards. Your body just continues ahead with the speed it has until you hit something. If you look outside at the ground, it doesn't look like you are pushed forward - but when you look at the bus around you, it does look like you are thrown forward. It is nothing but an illusion.
So, all this is actually quite simple. You are not accelerating out of or pushed out of the circle - your body is merely continuing with whichever speed it has at the moment. If you have a speed directed tangentially to a circle, then you are naturally moving out of that circle. Only if that circle holds on to you - like the merry-go-round rail - you will move in a circular path because it constantly pulls you back in and avoids that you fly out of the circle.
In short: There is a centre-directed force due to the car door or merry-go-round rail, which pulls you inwards (we call this a centripetal force). But there is no force pulling you outwards. This feeling of falling outwards can be called a centrifugal effect or so if you will - but avoid the term centrifugal force, because it really isn't a force.
Now, regarding your teacher talking about inertial frames of reference
From the duvet and bus examples above, it is clear that depending on from where you are standing and watching, it looks different - a camera on the bus or on the duvet sees you throw forwards, while a camera at the bus stop or at the table sees you with no change in motion (you either keep standing still on the table of keep continuing with your speed in the bus).
So the point, from where we look makes a big difference. We call this point a reference frame. And if you look at the two examples again, it seems that whenever the reference frame is accelerating (the pulled duvet, the braking bus, the turning car, the turning merry-go-roung), then this illusion appears.
So, in order to avoid this illusion, people in the past have invented the term inertial reference frame. We call a reference frame inertial, if it does not accelerate. Only then do we not see any illusions, and only then do Newton's laws hold true.