Backdoors simply allow you to start a Hacking Operation from that specific node/system. Meaning if you wanted to hack that same particular system again, you can do so. Placing backdoors is useful since they allow you to hack the enemy without placing your own systems at risk of retaliatory action from Defensive Hacking systems(Defensive Hacking Systems trace the source of a Hack). If they(Defensive Hacking Systems) are succesful in tracing you, they can effectively "Blackout" your system or prevent you from changing build queue and other actions. It also yields a small speed buff when running a Hack through the node(but why do that when you have a Backdoor?) and it prevents the enemy from conclusively knowing who's hacking them, although if you hack from a Backdoor, Defensive trace actions will destroy backdoors if discovered. Small price to pay compared to a Blackout as I see it.

When successfully hacking a system, you can place Sleepers. Sleepers are effectively Spies or Brainwashed citizens. Get enough Sleepers within an Empire and they provide bonuses(like Luxury and Strategic Resources). You can also alter system Politics, causing significant unrest since they're not represented in the current Government(although its not as strong as you might think). If a Governor is present, you can remove them from play for 20 turns(less if they have Hero Recovery buffs like the Vodyani). Finally you can Jam Commands. This basically prevents the player from issuing commands to the system's Ground Forces, making them less organized(-50% Defense Garrison Deployment) and unable to carry out useful Battle Tactics like Draft that sacrifices population to add to the system's garrison.

When hacking a Home/Capital System, more options are available. You can steal technology(Server ID, spawns a free Backdoor), Support Revolutionaries(Spawns Pirates and Jams Commands) and Alter their Government(basically throw their Government into Anarchy). Defense systems on your Home System and surrounding systems are highly advised.

Umbral Choir faction is built around hacking and I haven't played them yet so I can't really give you advice about them. However they have more options than other factions when it comes to hacking, including abducting Sleepers.

I accidentally discovered something nice about backdoors that I never realized before.  Maybe there was a patch change I missed?

Anyway, so normally when you cancel a hack it costs you a temporary 30 bandwidth, right?  Well, if you initiated your hack from a backdoor, you don't need to cancel the hack. Instead, just kill the backdoor and it cancels the hack for free. You actually end up recovering some bandwidth because every backdoor costs a little bit of bandwidth.

If you google hacking guides, you'll find some good stuff.  I think there are a few things about hacking that are not clear that I might help you with:

1) Defensive programs only go on friendly systems. ALL of these will initiate a trace, plus the other function(s) they list in their description. They are more valuable in FRONT of systems you want to protect than on the system itself (but sometimes you don't have that option of course).

2) Offensive programs only go on enemy systems. Also, they can only be run during an ACTIVE hack, unlike defensive programs which run "in the background" all the time. This can be a little counterintuitive, especially for something like "Accelerator" which I would love to be able to install in a friendly system! It's more about puzzle gameplay balance here than realism, but I think it's fun once you figure out the implications.  I have yet to play with hacking in MP; I think it would be fun trying to outhack an opponent.

3) Both defensive and offensive programs use bandwidth, so you'll never "have enough". You sort of have to choose whether to play more defensively or offensively, but you can switch it up on the fly. This plus the slow speed of hacking means that typically I find that I can only deal with one opponent at a time, so offensive hacking (against full factions) requires some forethought about what you want to get out of it. You could always just use it on pirates and minor factions though!

4) Cancelling your hack is necessary to stop a trace against you, but costs you a 30 bandwidth "feedback" for several turns! If you don't have that available, you have to shut down programs until you do, leaving yourself vulnerable. Keep this in mind any time you start a risky hack.

5) The wording on hacking programs is very poorly described in my opinion. Feel free to ask about any individual program for clarity (if google doesn't help you first). There are a few I still don't understand, but someone will hopefully come along that does.

4) Traces operate at +50% (or maybe +100%) speed.  This means you can calculate how long an enemy trace will take.  I strongly advise keeping this in mind.  It means that longer routes before you strike a defensive node are typically safer (12+ turn hacks) but it does depend on the nodes.

4) traces operate at 100% speed of the Empires hacking speed. If an Empire got 110 hacking speed his trace will be at 220, or 55 hacking speed on endless game speed he will have a trace speed of 110. 

What Im curious to know is how do you detect a backdoor and exactly how do you get rid of one once detected?

also, a backdoor gives your hack a 5% hacking speed bonus, which also applys to nearby starlanes/systems.