You really shouldn't use the "other" Wi-Fi channels, but here are some reasons for why they may be used, as well as some general information about 802.11 channels and interference.
When I talk about reliability, I'm referring to a wireless link that delivers a constant minimum speed, which is very important for things like VoIP and video conferencing. Speed refers to the average throughput which is important for downloads.
In the US, you can use channels 1 to 11 (or 1 to 9), giving you 3 non overlapping 22MHz (or 20MHz) channels, and in Europe channels 1 to 13 can be used, providing 4 non overlapping 20MHz channels, or two non interfering 40MHz N mode channels. Each channel is 5MHz wide and Wi-Fi needs 20MHz of separation. 11b DSSS/CCK Wi-Fi actually uses 22MHz, leading to the more ideal 25MHz recommended spacing of channels 1, 6, and 11. That's mostly obsolete, but even g networks fall back to DSSS at their lowest bitrates, so 25MHz still can help a little.
The 5GHz band has 9 non overlapping 20MHz channels (notice how they skip by 4), with the some of the newer equipment adding 4 or more channels.
Reason 1: All of your Wi-Fi client devices remain very close to your access point at all times, and you don't care about causing interference to others or having a reliable connection farther away. For example, you have neighbors with networks on channels 1, 6, and 11, but when doing a speed test while being very close to your access point, you found that using an in between channel such as channel 3 was the fastest. The reason is your wireless devices avoid generating interference by not transmitting when they can detect other Wi-Fi traffic being transmitted on the same channel. By using channel 3, this feature is effectively disabled and your devices can no longer see traffic from your neighbors' networks. Your devices then operate at full speed because no interference is detected. As long as your devices remain very close to your access point, the interference from your neighbors on channels 1 and 6 won't be strong enough to cause interference to you. But now the users on channels 1, 3 or 6 will have horrible reliability if they move farther away if two of the overlapping channels are in use at the same time.
Reason 2: You're using 11b DSSS modes that are more tolerant to overlapping. Because these are spread spectrum, a channel that overlaps somewhat just degrades the quality of the link resulting in a lower possible bit rate or range. You may be able to squeeze 4 channels in to the channel 1 to 11 range and obtain higher performance. 11b is long obsolete and there is really no reason to do this when you can have 3 non interfering 54mbps OFDM channels (or 4 in Europe). Have you ever seen your Wi-Fi card transmit at 2, 5.5, or 11mbps DSSS (11b) modes when 6mbps OFDM (11g) should provide better range than 2mbps DSSS? This may be because DSSS is more tolerant to a partially overlapping channel than OFDM is.
Reason 3: You're still using some very old wireless equipment that predates the 11b standard, or you're using a special narrow band 5MHz wireless channel, or you're trying to avoid interference from a narrow band device like a baby monitor or microwave oven. In this case you might use channels 1, 5, and 9 leaving the top end of the band (above channel 11) open for the other equipment.
Wi-Fi is meant to generate minimal interference when configured correctly. Each wireless frame contains a header which is broadcast at the slowest speed. It contains the preamble and the packet length. The high speed data follows after it. This is done so that all nodes in the area can receive the frame header and not transmit until that frame finishes broadcasting. When nodes are too far part to see each other's headers, the network switches in to RTS/CTS mode so that all nodes receive a signal from the access point to stay quiet while an out of range node is transmitting. This also applies to mixed 11b and 11g devices since 11b devices can't receive 11g frame headers. When an access point is set on an overlapping in between channel, all this falls apart.
A lot has changed in the 7 years since this question was posted. Cheap double channel width 11n devices have become common place. More recently, 11ac devices which can combine up to 8 of the 9 or more available channels to make a super wide high speed channel in the 5GHz band are becoming common place.
Unlike the older 108mbps Atheros hardware that uses the second channel only as needed and when it detects that it isn't busy, the new 11n standard doesn't have such good interference reduction. It operates at double wide channel mode all the time when 40MHz channel mode is enabled. It's so bad that most people completely disable the 40MHz N mode in any urban environment.
Some of the responses said to move to 5GHz. With 11ac becoming common place, it may no longer be so easy to even find a single (20MHz) channel to use if 4 or 8 channel wide 11ac is in use nearby. 11ac is supposed to be better at not generating interference on the bonded channels when they're already in use, but I don't know how well that works. Many of the 5GHz clients connecting to the new 11ac access points are actually b/g/a/n clients connecting in n mode, and they generate the same interference that n does on 2.4GHz.
If you want to increase your speed without generating and receiving more interference, it's best to use MiMO modes to get 2 or even 3 data streams out of a single 20MHz channel. Unfortunately the ultra compact mobile devices usually don't support multiple MiMO streams.
Improperly configured access points, cheap channel bonding access points without MiMO, and round the clock streaming have made Wi-Fi reliability far worse than it was 10 years ago. I hope this information helps.
Detailed information about the Wi-Fi frame format: