A set of N linear equations are dependent if one equation can be written as the linear combination of the remaining equations.
Any set of values that satisfying the first N-1 equations will also satisfy their linear combination - the Nth equation. So there is no point in adding an extra redundant equation to the set when N-1 equations can do the same job.
Consider the set of equations given in the original post. Equation (2.8) can be represented as:
$$(2.8) = -(2.5)-(2.6)-(2.7)$$
So any set of values of currents satisfying equations (2.5),(2.6) and (2.7) will also satisfy their linear combination - and hence equation (2.8). So adding equation (2.8) is adding redundancy.
Dependency expressed in other words:
A set of N equations are dependent if there exist any linear combination of these N equation that gives a result 0=0:
where \$c_i\ne 0\$. This definition is used in OP to say that the set of equations are dependent.