What is dBm?
dBm stands for decibels relative to one milliwatt. Decibels represent multiplicative factors, or ratios; by establishing a specific reference level they can instead be used as absolute values: 0 dBm is 1 milliwatt, 3 dBm is approximately 2 milliwatts, etc.
How do I convert it to watts?
Convert the decibel value to a scale factor and multiply by one milliwatt. That is,
$$x_{\mathrm{mW}} = 10^{x_{\mathrm{dBm}}/10} \cdot 1 \,\mathrm{mW}$$
$$x_{\mathrm{W}} = 10^{x_{\mathrm{dBm}}/10} \cdot 0.001 \,\mathrm{W}$$
For example, the datasheet you link mentions a value of
$$17 \,\mathrm{dBm} = 10^{17/10} \,\mathrm{mW} \approx 50.1 \,\mathrm{mW}$$
Why, or when, would you use dBm to specify power output rather than watts?
Are there specific problems or equations that are easier to deal with in dBm vs watts?
Gain and loss in all stages of an RF system (feed line, filters, amplifiers) is multiplicative (if it were not, that would be nonlinearity), and therefore is typically written in dB so that the total gain or loss may be computed by adding, rather than multiplying, all the individual values together.
If you add a value in dB to a value in dBm, the result is in dBm. (Adding two dBm values is not usually meaningful since it would correspond to power squared.)