what does a dot product indicate?

consider two vectors a=, b= x,y,p and q are real numbers representing the length of the components of the unit vectors in the x and y direction (here x and y refer to the cordinate axis not the x,y real numbers mentioned earlier) usually labelled i and j (This is actually a special case because i and j are orthogonal ie at right angles to one another. In the general case they need not be.)

what this means is that a = < x, y > = xi + yj  and b= = pi + qj
remember  x,y,p and q are scalar real numbers without direction, they contribute the magnitudes whilst i and j impart direction with just unit magnitude.

what is i.i ?  By definition IiI.IiI cos0 = 1 x 1 x1 = 1 = IjI.IjI cos0

what about i.j and j.i well because they are at right angles to each other and cos 90 = 0 they are both zero

so what about a.b?

(xi + yj).(pi + qj) = xp(i.i) + xq(i.j) + yp(j.i) + yq(j.j)

= xp(1) + xq(0) + yp(0) + yq(1) = xp + yq which is a real number since

x,y,p and q are all real numbers.

This shouldn't come as any surprise as you accepted

a.b=lal.lbl.cos theta. but this is also a real number

IaI is just a number representing the magnitude of a

IbI is just a number representing the magnitude of b

and cos theta is of course a number

therefore for a =  = xi + yj  and b = = pi + qj

a.b=lal.lbl.cos theta = xp + yq

now because IaI = sqrt(x^2 + y^2) and IbI = sqrt(p^2 + q^2) by Pythagoras theorem

lal.lbl.cos theta = xp + yq

cos theta = (xp + yq)/lal.lbl =

(xp + yq)/(sqrt(x^2 + y^2)*sqrt(p^2 + q^2)

which gives a direct means of calculating cos theta and hence theta the angle between the two vectors a and b

Your query is what does the real number xp+yq represent. Well first it is a scalar quantity and it represents the same quantity
a.b=lal.lbl.cos ?   i.e. xp + yq =  lal.lbl.cos ?
In the first expression you know the magnitude of the vectors a and b and the angle ? between them . In the second representation you know the components of the vectors a and b in the orthogonal ( i , j , k ) directions.

One physical example of a dot product that comes to mind is the dot product of force and displacement ( both are vectors please note ) and which represents the work done by the force which is a scalar quantity.
Hence the dot product of two vector is always a scalar. Hope it is clear.