Why/How are there infinite points in a line segment?

Let's assume that your end points are A(0,0) and B(1,0).

Try to find a point in between them. Ok you found C(0.4,0).

Now try finding another point. Will you ever NOT be able to find another point? No, because there are infinite numbers between 0 and 1 (just think of decimals).

In the same way there are infinite points in a line segment.

Any segment with at least two points has infinitely many points, because, intuitively, given any two distinct points, there is a third one, distinct from both of them, say, the middle point.

If there were a finite number of points, there will be no point between say point 6 and point 7 (etc)

The concept of infinity is used for many different things, and one should not confuse them. A line, be it closed, open, straight, curved, finite in length, or infinite in length always consists of infinitely many individual points (at least for any reasonable notion of 'line'). This can be proven rigorously, and it's not hard at all. Basically, between any two distinct points on a line there is a third point between these two points (between should not necessarily mean the mid-point on a straight path connecting the points, and this may get a bit tricky, but not too tricky). The length of the line is a different issue. It may be finite in length or infinite in length. The totality of the points comprising the line is in any case infinite. In fact it can be shown (quite easily) that the cardinality of points of any line is always the same, so in a sense all lines have the same "amount" of points in them, though the way these points are arranged may give totally different geometric qualities to the line.