Carrier bands are analogous to TV or radio station "channels." Each carrier operates a specified network technology and are assigned bands in which they can operate. In order for your phone to connect to a carrier's network, your phone must be able to connect to at least one of the carrier's bands. The more bands, the more ways to connect to the carrier network.
Depending on how many of that carrier's bands you can connect to determines the quality and availability of the service. It doesn't necessarily mean that you'll have poor connections with the carrier, but rather you may not get signal in a particular market. Whether or not you get dropped calls or poor service has more to do with the capacity, saturation, and signal strength than the number of different bands.
For example, you might be able to get excellent coverage in the city, but if you go to the outskirts of the suburbs or into a rural area, you may lose coverage. They may be using band 66 (this is an arbitrary example and not necessarily the actual case) out in the farmland or countryside. Since your phone doesn't support band 66, you won't be able to use this band to access the network.