Let this différential equation : $$(E) :\quad ay"+by'+cy=d(t)$$
And :$$(H) :\quad ay"+by'+cy=0$$
We suppose that $f(x)=\lambda_1f_1(x)+\lambda_2f_2(x)$ is the general solution of $(H)$
The method of variation of constants says that we have to solve this system :
$$\left\lbrace\begin{array}{l}\lambda_1(x)f_1(x)+\lambda_2(x)f_2(x)=f(x)\\\\\lambda_1(x)f_1'(x)+\lambda_2(x)f_2'(x)=f'(x)\end{array}\right.$$
But $f'(x)=\color{red} {\lambda_1'(x)f_1(x)+\lambda_2'(x)f_2(x)}+ \lambda_1(x)f_1'(x)+\lambda_2(x)f_2'(x)$
Why we must suppose that the part in red is nul ($\lambda_1'(x)f_1(x)+\lambda_2'(x)f_2(x)=0$)
You have way too many typos/mistakes in what you have typed, so I will use my own notation.
Consider the second order ODE $$ y''+ay'+by=f. $$ Assume that $y_1,y_2$ are two linearly independent solutions of the homogeneous equation, and hence the general solution of the homogeneous equation is $$ y_h(t)=C_1y_1(t)+C_2y_2(t). $$ Now you are using the method of the variation of the parameter to end up with the system $$ C_1'y_1+C_2'y_2=0,\\ C_1'y_1'+C_2'y_2'=f,\tag{$\ast$} $$ where the first equation is a rather arbitrary and mysterious assumption that is necessary to make to have the second equation. It is a very good question to ask "how so" in this situation.
The short answer is because it works.
The longer answer needs a little more theory. In particular, the original equation is equivalent to the system of two first order equation (here $x_1=y,x_2=y'$) $$ x_1'=x_2,\\ x_2'=-bx_1-ax_2+f, $$ with the fundamental matrix $$ \Phi=\begin{bmatrix} y_1&y_2\\ y_1'&y_2'\\ \end{bmatrix} $$ It means that the general solution to the homogeneous system can be written as $$ x(t)=\Phi(t)C,\quad C=(C_1,C_2). $$ Method of the variation of the parameter, i.e., assuming that $C$ is not a constant but depends on $t$, leads to $$ \Phi(t) C'(t)=F(t),\quad F(t)=(0,f(t)), $$ which is exactly the system $(\ast)$, without any mysterious assumptions.
So, the longer answer is that theoretically it is correct to look at this equation as at a system, and use the methods that work for the systems. These methods may be a little too advanced for many undergraduates learning ODE, and hence we adapt them. These adaptations usually lead to a lot of handwaving (cf. with modern Calculus textbooks).