How does modulation work in simple language?
In order to understand the integral occurring in frequency modulation you need to understand the relation between phase and frequency. A sinusoid with frequency $f$ can be written as

$$x(t)=\sin(2\pi f t)\tag{1}$$

where the argument $2\pi f t$ is called the phase $\phi(t)$. The frequency is the derivative of the phase divided by $2\pi$:

$$f=\frac{1}{2\pi}\frac{d\phi(t)}{dt}\tag{2}$$

If the frequency is not constant, the relation $(2)$ still holds. Assume now that the frequency is a function of time $f(t)$. From ($2$), the phase can be determined from the frequency by integration:

$$\phi(t)=2\pi\int_{0}^tf(\tau)d\tau\tag{3}$$

In frequency modulation, the frequency of the carrier is changed according to the message signal $m(t)$ around a given (average) carrier frequency $f_c$:

$$f(t)=f_c+\Delta f\cdot m(t)\tag{4}$$

where $\Delta f$ is the maximum frequency deviation, assuming that $|m(t)|\le 1$. From $(3)$ and $(4)$ we can obtain the phase

$$\phi(t)=2\pi f_ct+2\pi\Delta f\int_0^tm(\tau)d\tau\tag{5}$$

From $(5)$, the frequency modulated signal can be written as

$$s(t)=\sin(\phi(t))=\sin\left(2\pi f_ct+2\pi\Delta f\int_0^tm(\tau)d\tau\right)\tag{6}$$



Your understanding of amplitude modulation seems to be correct. You simply multiply the carrier with a non-negative signal that is obtained from the message signal plus a constant. The constant must be chosen such that the sum remains positive.