How do I calculate what resistors I need for RGB LEDs with 3.7V?
It is recommended to have a max of 20mA per color. Assume there is an ideal voltage drop of 2V, 3.2V, and 3.2V across the red, green, and blue LED's respectively (data from the linked datasheet).

The theoretical minimum resistor values you can "safely" use are:
$$
R_{red} = \frac{3.7V - 2V}{20mA} = 85 \Omega
$$
$$
R_{blue} = R_{green} = \frac{3.7V - 3.2V}{20mA} = 25 \Omega
$$

Usually in practice you'll have perfectly visible LED's even with 10mA per channel, in which case simply double the resistances.

The choice of \$100\Omega\$ and \$150\Omega\$ seem arbitrary, vaguely taking into account the fact that a higher resistance is needed for the red LED, but not quite scaling at the right proportion to get an equivalent current in each channel or any other useful balance I can tell. They're perfectly safe to use, and in practice probably won't make a difference.