A first-order linear ODE has the form

$$y'+p(x)y=q(x).$$

Its defining feature is that $y$ and $y'$ appear only to the first power and are not multiplied together.

An integrating factor

$$\mu(x)=e^{\int p(x),dx}$$

turns the left side into a product derivative:

$$\mu y'+\mu py=(\mu y)'.$$

Therefore

$$(\mu y)'=\mu q,$$

so integration gives

$$\mu(x)y(x)=\int \mu(x)q(x),dx+C.$$

The method works because the integrating factor is chosen to manufacture the product rule. Initial conditions determine the constant after the general solution is obtained.