When the index of a Tensor is repeated $a_{\beta j}a_{\beta l} = \delta_{jl}$ is the dot product of columns j and l of rotation matrix A.
But This dot product can have only two values 1 or 0 and replaced by Kronecker Delta Function $\delta_{jl}$
Why this dot product is defined in this way so that in can have only 2 values viz. 1 or 0 and not in between?
It seems that $a$ is a real operator representing a rotation about the origin in $\mathbb R^n$, and furthermore that the indices are with respect to some orthonoral basis $\vec e_i$ (index notation is also used in non-orthonormal bases, but in that case it is typical to distinguish between covariant and controvariant indices, which as far as I can tell isn't the case here). You might want to look through the the text and verify that this is indeed the case.
Rotations have a few useful properties, chief among which is that they are inner product preserving (often, this is used as the definition of a rotation): $$\langle a\vec u,a\vec v\rangle = \langle\vec u,\vec v\rangle\ \ \ \forall \vec u,\vec v$$ Of course, the columns of $a$ are precisely the results of $a$ acting on the standard basis vectors $\vec e_i$, from which we see that if the basis vectors are orthonormal, so too are their images under $a$: $$\langle a\vec e_j,a\vec e_l\rangle = \langle\vec e_j,\vec e_l\rangle=\delta_{jl}$$ In index notation, that is: $$a_{\beta j}a_{\beta l} = \delta_{jl}$$